%%Page: 1 1 TeXDict begin HPSdict begin 1 0 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 0 a SDict begin [ /Producer (dvips + Distiller) /Title () /Subject () /Creator (LaTeX with hyperref package) /Author () /Keywords () /DOCINFO pdfmark end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.1) cvn H.B /DEST pdfmark end -913 1926 a Black Black -913 3033 a SDict begin [ /Count -3 /Dest (chapter.1) cvn /Title (To Begin) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -2 /Dest (section.1.1) cvn /Title (General Information) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.1.1.1) cvn /Title (Starting the Calculator) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.1.1.2) cvn /Title (What You See) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.1.2) cvn /Title (Notations) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.1.3) cvn /Title (Online Help) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -4 /Dest (chapter.2) cvn /Title (Aplets) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.2.1) cvn /Title (APLET Key) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.2.2) cvn /Title (The Various aplets) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.2.3) cvn /Title (Examples using the Sequence Aplet) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.2.3.1) cvn /Title (Notation in Base b) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.2.3.2) cvn /Title (Calculating the GCD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.2.3.3) cvn /Title (Calculating the Coefficients of B\351zout's Identity) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.2.4) cvn /Title (The SYMB, NUM, PLOT and MODES Keys) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -4 /Dest (chapter.3) cvn /Title (CAS user's guide) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.3.1) cvn /Title (Terms and Conditions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -4 /Dest (section.3.2) cvn /Title (Introduction) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.2.1) cvn /Title (What is CAS?) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.2.2) cvn /Title (How do you perform a symbolic calculation?) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.2.3) cvn /Title (Variables) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.2.4) cvn /Title (The current variable) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -5 /Dest (section.3.3) cvn /Title (Using the CAS in the Equation Editor) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.3.1) cvn /Title (The Equation editor menu) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.3.2) cvn /Title (The keyboard in the Equation Editor) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.3.3) cvn /Title (Expressions, subexpressions, selection) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.3.4) cvn /Title (Accessing the CAS functions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.3.5) cvn /Title (Equation editor variables) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -6 /Dest (section.3.4) cvn /Title (CAS in the HOME screen) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.4.1) cvn /Title (HOME variables) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.4.2) cvn /Title (Storing an object in a HOME variable : STO) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.4.3) cvn /Title (Some tricks and examples.) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.4.4) cvn /Title (Send expressions from HOME into the CAS history : PUSH) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.4.5) cvn /Title (Retrieve expressions written to the CAS history in HOME : POP) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.3.4.6) cvn /Title (The keyboard in the HOME screen) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -13 /Dest (chapter.4) cvn /Title (The CAS functions only valid in HOME screen) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -2 /Dest (section.4.1) cvn /Title (Linear systems) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.1.1) cvn /Title (Resolution of a linear system of equations : REF) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.1.2) cvn /Title (Resolution of a linear system of equations : rref) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -11 /Dest (section.4.2) cvn /Title (Polynomials) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.1) cvn /Title (List of factors : FACTORS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.2) cvn /Title (Rational function given by its roots and poles :FCOEF) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.3) cvn /Title (Roots and poles ofa rational function : FROOTS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.4) cvn /Title (Gr\234bner basis : GBASIS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.5) cvn /Title (Gr\234bner reduction : GREDUCE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.6) cvn /Title (Lagrange's polynomial : LAGRANGE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.7) cvn /Title (GCD of a list : LGCD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.8) cvn /Title (resultant of two polynomials: RESULTANT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.9) cvn /Title (Division by the PGCD :SIMP2) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.10) cvn /Title (Sturm's sequences : STURM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.2.11) cvn /Title (Zeros of a polynomial : ZEROS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -1 /Dest (section.4.3) cvn /Title (Rational fractions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.3.1) cvn /Title (Numerator and denominator of a fraction : FXND) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -1 /Dest (section.4.4) cvn /Title (Modular calculations) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.4.1) cvn /Title (Resolution of a linear system of equations : RREFMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -2 /Dest (section.4.5) cvn /Title (Variables) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.5.1) cvn /Title (The symbolic variables of an expression : LNAME) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.5.2) cvn /Title (The variables of an expression : LVAR) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -5 /Dest (section.4.6) cvn /Title (Numeric calculus) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.6.1) cvn /Title (Omit the values smaller than EPS : EPSX0) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.6.2) cvn /Title (Solve numerically a system : MSLV) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.6.3) cvn /Title (Time for evaluation of the answer : TEVAL) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.6.4) cvn /Title (Numeric approximation : XNUM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.6.5) cvn /Title (Exact value : XQ) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -2 /Dest (section.4.7) cvn /Title (The lists) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.7.1) cvn /Title (Concatenation : AUGMENT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.7.2) cvn /Title (Map a function to a list : MAP) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -4 /Dest (section.4.8) cvn /Title (Quadratic forms) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.8.1) cvn /Title (Matrice to quadratic form : AXQ) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.8.2) cvn /Title (Gauss decompsition : GAUSS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.8.3) cvn /Title (Matrix of a quadratic form : QXA) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.8.4) cvn /Title (Sylvester decomposition : SYLVESTER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -5 /Dest (section.4.9) cvn /Title (Functions of multiple variables) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.9.1) cvn /Title (The rotor : CURL) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.9.2) cvn /Title (The gradient : DERIV) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.9.3) cvn /Title (The divergence : DIV) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.9.4) cvn /Title (The hessian : HESS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.9.5) cvn /Title (The laplacian : LAPL) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -1 /Dest (section.4.10) cvn /Title (Equations) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.10.1) cvn /Title (Members of an equation : EXLR) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -19 /Dest (section.4.11) cvn /Title (Matrices) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.1) cvn /Title (Matrix to list : AXL) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.2) cvn /Title (Symbolic matrix to numeric matrix : AXM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.3) cvn /Title (Basis vectors : BASIS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.4) cvn /Title (Cholesky matrix : CHOLESKY) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.5) cvn /Title (Map an operator to a diagonalizable matrix : DIAGMAP) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.6) cvn /Title (Hadamard product : HADAMARD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.7) cvn /Title (Hilbert matrix : HILBERT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.8) cvn /Title (Basis of an intersection : IBASIS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.9) cvn /Title (basis of the image : IMAGE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.10) cvn /Title (Jordan matrix : JORDAN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.11) cvn /Title (Basis of the kernel : KER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.12) cvn /Title (Make a matrix : LCXM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.13) cvn /Title (List of several elements : MAD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.14) cvn /Title (Characteristic polynomial : PCAR) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.15) cvn /Title (Minimal polynomial : PMINI) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.16) cvn /Title (Q*R decomposition : qr) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.17) cvn /Title (System to matrix : SYST2MAT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.18) cvn /Title (Transposed matrix : TRAN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.11.19) cvn /Title (Vandermonde matrix : VANDERMONDE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.4.12) cvn /Title (The permutations) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.12.1) cvn /Title (Cycles to permutation : C2P) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.12.2) cvn /Title (Composition : CIRC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.12.3) cvn /Title (Decomposition into cyclesP2C) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -2 /Dest (section.4.13) cvn /Title (The isometries) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.13.1) cvn /Title (Isometry's characteristics elements : ISOM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.4.13.2) cvn /Title (Isometry's matrix : MKISOM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -16 /Dest (chapter.5) cvn /Title (CAS Reference from the Equation Editor) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -9 /Dest (section.5.1) cvn /Title (CAS menu bar) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.1) cvn /Title (Change your configuration : CFG) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.2) cvn /Title (Step/Step Mode) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.3) cvn /Title (TOOL menu) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.4) cvn /Title (ALGB menu) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.5) cvn /Title (DIFF menu) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.6) cvn /Title (REWRI menu) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.7) cvn /Title (SOLV menu) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.8) cvn /Title (TRIG menu) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.1.9) cvn /Title (The MATH key) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -17 /Dest (section.5.2) cvn /Title (Integers \(and Gaussian Integers\)) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.1) cvn /Title (Unlimited precision) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.2) cvn /Title (The divisors of a number : DIVIS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.3) cvn /Title (The Euler index : EULER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.4) cvn /Title (Decomposition into prime factors : FACTOR ) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.5) cvn /Title (The greatest common divisor : GCD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.6) cvn /Title (Resolution of au+bv=c : IABCUV) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.7) cvn /Title (Chinese remainders :ICHINREM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.8) cvn /Title (Quotient and remainder of two integers : IDIV2) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.9) cvn /Title (B\351zout's Identity : IEGCD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.10) cvn /Title (The integer Euclidean quotient : IQUOT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.11) cvn /Title (The integer remainder : IREMAINDER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.12) cvn /Title (Test of pseudo-primality : ISPRIME?) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.13) cvn /Title (The least common multiple : LCM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.14) cvn /Title (Remainder of the Euclidean division : MOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.15) cvn /Title (The smallest pseudo-prime greater than N : NEXTPRIME) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.16) cvn /Title (Resolution of a2+b2=p with integers: PA2B2) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.2.17) cvn /Title (The greatest pseudo-prime less than N : PREVPRIME) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -11 /Dest (section.5.3) cvn /Title (Modular Calculations) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.1) cvn /Title (Addition in Z/pZ or in Z/pZ[X] : ADDTMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.2) cvn /Title (Quotient and remainder in Z/pZ[X]: DIV2MOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.3) cvn /Title (Division in Z/pZ or in Z/pZ[X] : DIVMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.4) cvn /Title (To expand and simplify expressions in \040Z/pZ or in Z/pZ[X] : EXPANDMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.5) cvn /Title (To factor polynomials in Z/pZ[X] : FACTORMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.6) cvn /Title (GCD of the two polynomials in Z/pZ[X] : GCDMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.7) cvn /Title (Inverse in Z/pZ : INVMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.8) cvn /Title (To put a value in the MODULO variable : MODSTO) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.9) cvn /Title (Multiplication in Z/pZ or Z/pZ[X] : MULTMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.10) cvn /Title (Power in Z/pZ or in Z/pZ[X] : POWMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.3.11) cvn /Title (Subtraction in Z/pZ or in Z/pZ[X] : SUBTMOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.5.4) cvn /Title (Rational Numbers) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.4.1) cvn /Title (The n-th Bernoulli's number : IBERNOULLI) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.4.2) cvn /Title (Whole-number and fractionnal part : PROPFRAC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.4.3) cvn /Title (Symbolic summation : ) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -5 /Dest (section.5.5) cvn /Title (Real Numbers) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.5.1) cvn /Title (The whole part : FLOOR) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.5.2) cvn /Title (The \040function : GAMMA) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.5.3) cvn /Title (Remainder of the Euclidean division : MOD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.5.4) cvn /Title (The derivatives of the Digamma function :PSI) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.5.5) cvn /Title (The Digamma function : Psi) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.5.6) cvn /Title (Complex Numbers) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.6.1) cvn /Title (Argument of a complex number : ARG) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.6.2) cvn /Title (Conjugate of a complex number : CONJ) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.6.3) cvn /Title (Equation of the line : DROITE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -7 /Dest (section.5.7) cvn /Title (Algebraic Expressions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.7.1) cvn /Title (Factors over the integers : COLLECT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.7.2) cvn /Title (Distributivity : EXPAND) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.7.3) cvn /Title (Factorization : FACTOR) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.7.4) cvn /Title (Change the order of variables : REORDER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.7.5) cvn /Title (Partial simplifications : SEVAL) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.7.6) cvn /Title (Substitute a value for a variable : |) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.7.7) cvn /Title (Substitute a value for a variable : SUBST) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -19 /Dest (section.5.8) cvn /Title (Polynomials) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.1) cvn /Title (Resolution of A[X]U[X]+B[X]V[X]=C[X] : ABCUV) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.2) cvn /Title (Chinese remainders : CHINREM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.3) cvn /Title (Cyclotomic polynomial : CYCLOTOMIC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.4) cvn /Title (degree of a polynomial : DEGREE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.5) cvn /Title (Quotient and remainder : DIV2) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.6) cvn /Title (B\351zout's Identity : EGCD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.7) cvn /Title (Factorize a polynomial : FACTOR) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.8) cvn /Title (GCD of two polynomials : GCD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.9) cvn /Title (Hermite polynomial : HERMITE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.10) cvn /Title (To evaluate a polynomial : HORNER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.11) cvn /Title (LCM of two polynomials : LCM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.12) cvn /Title (Legendre polynomial : LEGENDRE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.13) cvn /Title (Partial fraction expansion : PARTFRAC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.14) cvn /Title (Whole-number and fractional part : PROPFRAC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.15) cvn /Title (Rewrite in order of the powers of \(X-a\) : PTAYL) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.16) cvn /Title (Euclidean quotient: QUOT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.17) cvn /Title (Euclidean remainder : REMAINDER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.18) cvn /Title (Number of zeros in [a,b[ : STURMAB) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.8.19) cvn /Title (First and 2nd-species Tchebycheff polynomials : TCHEBYCHEFF) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -15 /Dest (section.5.9) cvn /Title (Functions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.1) cvn /Title (Define a function : DEF) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.2) cvn /Title (Derivative and partial derivative : DERIV) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.3) cvn /Title (Derivative : DERVX) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.4) cvn /Title (Fourier coefficients : FOURIER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.5) cvn /Title (Partial integration : IBP) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.6) cvn /Title (Integrals : ) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.7) cvn /Title (Primitive and definied integral : INTVX) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.8) cvn /Title (To calculate limits : LIMIT or lim) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.9) cvn /Title (Plot of an expression : PLOT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.10) cvn /Title (Add the plot of an expression :PLOTADD) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.11) cvn /Title (Evaluate a primitive : PREVAL) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.12) cvn /Title (Primitive and definied int\351gral : RISCH) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.13) cvn /Title (Discrete antiderivative : SIGMA) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.14) cvn /Title (Discrete antiderivative : SIGMAVX) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.9.15) cvn /Title (Variation table : TABVAR) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -4 /Dest (section.5.10) cvn /Title (Taylor and asymptotic Expansions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.10.1) cvn /Title (Division in increasing order by exponent : DIVPC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.10.2) cvn /Title (Limited nth-order expansion : SERIES) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.10.3) cvn /Title (Limited expansion in the vicinity of 0 : TAYLOR0) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.10.4) cvn /Title (Truncate at order n-1 : TRUNC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -11 /Dest (section.5.11) cvn /Title (Conversion Functions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.1) cvn /Title (Distributivity of multiplication : DISTRIB) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.2) cvn /Title (Disregard small values : EPSX0) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.3) cvn /Title (Transform exp\(n*ln\(x\)\) as a power of x : EXP2POW) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.4) cvn /Title (Transform a trigonometric expression into complex exponentials : EXPLN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.5) cvn /Title (Distributivity : FDISTRIB) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.6) cvn /Title (Linearise the exponentials : LIN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.7) cvn /Title (Regroup the logarithms : LNCOLLECT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.8) cvn /Title (Transform a power : POWEXPAND) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.9) cvn /Title (Simplify : SIMPLIFY) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.10) cvn /Title (Evaluation of reals : XNUM) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.11.11) cvn /Title (Rational approximation : XQ) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.5.12) cvn /Title (Equations) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.12.1) cvn /Title (The zeros of an expression :ISOLATE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.12.2) cvn /Title (Solve equations : SOLVE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.12.3) cvn /Title (Solve equations : SOLVEVX) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -2 /Dest (section.5.13) cvn /Title (Systems) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.13.1) cvn /Title (Solve linear system : LINSOLVE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.13.2) cvn /Title (Solve non linear system : SOLVE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.5.14) cvn /Title (Differential Equations) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.14.1) cvn /Title (Solve of differential equations : DESOLVE) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.14.2) cvn /Title (Laplace transform and inverse Laplace transform : LAP and ILAP) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.14.3) cvn /Title (Linear differential equations having constant coefficients : LDEC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -18 /Dest (section.5.15) cvn /Title (Trigonometric Expressions) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.1) cvn /Title (Transform the arccos into arcsin : ACOS2S) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.2) cvn /Title (Transform the arcsin into arccos : ASIN2C) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.3) cvn /Title (Transform the arcsin into arctan : ASIN2T) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.4) cvn /Title (Transform the arctan into arcsin : ATAN2S) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.5) cvn /Title (Transform exponential into hyperbolic functions : EXP2HYP) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.6) cvn /Title (Transform in terms of tan\(x/2\) : HALFTAN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.7) cvn /Title (Transform the complex exponentials into sin and cos : SINCOS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.8) cvn /Title (Transform tan\(x\) with sin\(2x\) and cos\(2x\) : TAN2CS2) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.9) cvn /Title (Replace tan\(x\) with sin\(x\)/cos\(x\) : TAN2SC) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.10) cvn /Title (Transform tan\(x\) with sin\(2x\) and cos\(2x\) : TAN2SC2) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.11) cvn /Title (Reconstruct the sine and the cosine of the same angle : TCOLLECT) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.12) cvn /Title (Develop transcendental expressions : TEXPAND) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.13) cvn /Title (Linearise une trigonometric expression : TLIN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.14) cvn /Title (Simplify using sin\(x\)2+cos\(x\)2=1 : TRIG) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.15) cvn /Title (Simplify using the cosines : TRIGCOS) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.16) cvn /Title (Simplify using the sines : TRIGSIN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.17) cvn /Title (Simplify using the tangents : TRIGTAN) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.5.15.18) cvn /Title (Simplification with complex exponentials : TSIMP) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.5.16) cvn /Title (Version number of your CAS : VER) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -4 /Dest (chapter.6) cvn /Title (Worked Exercises with the HP40) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.6.1) cvn /Title (Introduction) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -5 /Dest (section.6.2) cvn /Title (Exercises on Brevet) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.2.1) cvn /Title (Exercise 1) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.2.2) cvn /Title (Exercise 2) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.2.3) cvn /Title (Exercise 3) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.2.4) cvn /Title (Exercise 4) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.2.5) cvn /Title (Exercise 5) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.6.3) cvn /Title (Exercises on Bac) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.3.1) cvn /Title (Exercise 1) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.3.2) cvn /Title (Exercise 2 \(specialty\)) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.6.3.3) cvn /Title (Exercise 3 \(non-specialty\)) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (section.6.4) cvn /Title (Conclusion) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -14 /Dest (chapter.7) cvn /Title (Programming) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -4 /Dest (section.7.1) cvn /Title (Implementation) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.1.1) cvn /Title (How to edit and save a program) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.1.2) cvn /Title (How to correct a program) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.1.3) cvn /Title (How to run a program) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.1.4) cvn /Title (How to modify a program) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -1 /Dest (section.7.2) cvn /Title (Comments) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.2.1) cvn /Title (SHIFT 1 \(PROGRAM\) keys) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -3 /Dest (section.7.3) cvn /Title (Variables) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.3.1) cvn /Title (Their names) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.3.2) cvn /Title (The concept of local variables) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.3.3) cvn /Title (The concept of parameters) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -2 /Dest (section.7.4) cvn /Title (User entry) /OUT pdfmark end -913 3033 a -913 3033 a SDict begin [ /Count -0 /Dest (subsection.7.4.1) cvn /Title (Algorithmic 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x FF(4.7)1593 b(The)399 b(lists)-913 51642 y SDict begin H.S end -913 51642 a -913 51642 a SDict begin 12 H.A end -913 51642 a -913 51642 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.7.1) cvn H.B /DEST pdfmark end -913 51642 a 1530 x FE(4.7.1)1329 b(Concatenation)332 b(:)412 b Fq(AUGMENT)-913 55248 y FJ(AUGMENT)329 b FL(has)g(as)g(ar)-20 b(guments)330 b(tw)-11 b(o)328 b(v)-17 b(ectors,)343 b(or)328 b(tw)-11 b(o)329 b(lists,)339 b(or)329 b(a)g(list)e(and)j(an)f (ele-)-913 56576 y(ment.)p Black Black eop end end %%Page: 52 52 TeXDict begin HPSdict begin 52 51 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.52) cvn H.B /DEST pdfmark end -913 1926 a Black FL(52)p FD(CHAPTER)213 b(4.)532 b(THE)211 b FJ(CAS)g FD(FUNCTIONS)h(ONL)-111 b(Y)212 b(V)-149 b(ALID)210 b(IN)g(HOME)h(SCREEN)p Black -913 4140 a FJ(AUGMENT)278 b FL(concatenates)i(its)c(ar)-20 b(guments.)-913 5468 y(T)-89 b(yping:)11240 7270 y FA(AUGMENT)p Fw(\()p FI(f)p FA(1)p Fz(;)184 b FA(2)p FI(g)p Fz(;)g FA(3)p Fw(\))-913 9737 y FL(gi)-28 b(v)-17 b(es:)13705 11539 y FI(f)p FA(1)p Fz(;)184 b FA(2)p Fz(;)g FA(3)p FI(g)-913 14005 y FL(T)-89 b(yping:)10150 15807 y FA(AUGMENT)p Fw(\()p FI(f)p FA(1)p Fz(;)184 b FA(2)p FI(g)p Fz(;)g FI(f)p FA(3)p Fz(;)g FA(4)p FI(g)p Fw(\))-913 18274 y FL(gi)-28 b(v)-17 b(es:)13168 20076 y FI(f)p FA(1)p Fz(;)184 b FA(2)p Fz(;)g FA(3)p Fz(;)g FA(4)p FI(g)-913 22682 y SDict begin H.S end -913 22682 a -913 22682 a SDict begin 12 H.A end -913 22682 a -913 22682 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.7.2) cvn H.B /DEST pdfmark end -913 22682 a 1431 x FE(4.7.2)1329 b(Map)331 b(a)i(function)e(to)i(a)f(list)h(:)412 b Fq(MAP)-913 26489 y FJ(MAP)442 b FL(has)h(tw)-11 b(o)443 b(ar)-20 b(guments:)676 b(a)443 b(list)e(and)j(a)f(function)h(de\002ning)h(an)e (holomorphic)-913 27817 y(operator)-61 b(.)-913 29146 y FJ(MAP)277 b FL(applies)h(the)f(operator)h(to)f(the)h(list,)d(and)j (returns)f(the)g(result.)-913 30474 y(T)-89 b(yping:)11042 32276 y FA(MAP)p Fw(\()p FA(1)p Fz(;)184 b FA(0)p Fz(;)g FA(2)s Fz(;)g FA(EXP)p Fw(\()p FA(S1)p Fw(\))-913 34742 y FL(gi)-28 b(v)-17 b(es:)9798 36545 y FI(f)p FA(EXP)p Fw(\()p FA(1)p Fw(\))p Fz(;)184 b FA(EXP)p Fw(\()p FA(0)p Fw(\))p Fz(;)g FA(EXP)p Fw(\()p FA(2)p Fw(\))p FI(g)-913 39275 y SDict begin H.S end -913 39275 a -913 39275 a SDict begin 12 H.A end -913 39275 a -913 39275 a SDict begin [ /View [/XYZ H.V] /Dest (section.4.8) cvn H.B /DEST pdfmark end -913 39275 a 1874 x FF(4.8)1593 b(Quadratic)399 b(f)-40 b(orms)-913 42714 y SDict begin H.S end -913 42714 a -913 42714 a SDict begin 12 H.A end -913 42714 a -913 42714 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.8.1) cvn H.B /DEST pdfmark end -913 42714 a 1431 x FE(4.8.1)1329 b(Matrice)332 b(to)g(quadratic)g(f)-33 b(orm)331 b(:)413 b Fq(AXQ)-913 46520 y FJ(AXQ)391 b FL(has)g(tw)-11 b(o)391 b(ar)-20 b(guments:)572 b(a)391 b(symmetric)g(matrix)g Fz(A)g FL(representing)h(a)g(quadratic)-913 47849 y(form)276 b Fz(q)317 b FL(and)278 b(a)f(v)-17 b(ector)279 b(whose)f(components)h (are)f(the)f(quadratic)i(form')-61 b(s)276 b(v)-28 b(ariables.)-913 49177 y FJ(AXQ)277 b FL(returns)g(the)g(quadratic)i(form)d Fz(q)40 b FL(.)-913 50505 y(T)-89 b(yping)279 b(:)9259 52308 y FA(AXQ)p Fw(\([[)p FA(0)p Fz(;)184 b FA(1)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(0)p Fw(]])284 b Fz(;)461 b Fw([)p FA(S1)p Fz(;)184 b FA(S2)p Fw(]\))-913 54774 y FL(gi)-28 b(v)-17 b(es)278 b(:)13124 56576 y FA(2)246 b FI(\003)g FA(S2)h FI(\003)f FA(S1)p Black Black eop end end %%Page: 53 53 TeXDict begin HPSdict begin 53 52 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.53) cvn H.B /DEST pdfmark end -913 1926 a Black FD(4.8.)620 b(Q)-11 b(U)-44 b(ADRA)-123 b(TIC)278 b(FORMS)19317 b FL(53)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.8.2) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(4.8.2)1329 b(Gauss)331 b(decompsition)g(:)412 b Fq(GAUSS)-913 6262 y FJ(GAUSS)277 b FL(has)f(tw)-11 b(o)277 b(ar)-20 b(guments:)344 b(a)276 b(quadratic)j(form)d Fz(q)316 b FL(and)277 b(a)g(v)-17 b(ector)277 b(whose)h(compo-)-913 7590 y(nents)f(are)g(the)h(quadratic)h(form')-61 b(s)276 b(v)-28 b(ariables.)-913 8919 y FJ(GAUSS)277 b FL(returns)g Fz(q)317 b FL(e)-17 b(xpressed)278 b(as)f(a)g(sum)h(of)e(squares.)-913 10247 y(T)-89 b(yping:)9248 11649 y FA(GAUSS)p Fw(\()p FA(2)247 b FI(\001)g FA(S1)f FI(\001)g FA(S2)278 b Fz(;)461 b Fw([)p FA(S1)p Fz(;)184 b FA(S2)p Fw(]\))-913 13715 y FL(gi)-28 b(v)-17 b(es:)7863 15565 y FI(\000)p FA(2)246 b FI(\001)h Fw(\()10668 14816 y FA(S2)g FI(\000)e FA(S1)p 10668 15310 3678 45 v 12216 16324 a(2)14479 15565 y Fw(\))14909 15108 y Fo(2)15622 15565 y Fw(+)16862 14816 y FA(1)p 16862 15310 582 45 v 16862 16324 a(2)17822 15565 y FI(\001)h Fw(\()p FA(S2)h Fw(+)f FA(S1)p Fw(\))22913 15108 y Fo(2)-913 18053 y SDict begin H.S end -913 18053 a -913 18053 a SDict begin 12 H.A end -913 18053 a -913 18053 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.8.3) cvn H.B /DEST pdfmark end -913 18053 a 1051 x FE(4.8.3)1329 b(Matrix)332 b(of)h(a)f(quadratic)g (f)-33 b(orm)331 b(:)412 b Fq(QXA)-913 21226 y FJ(QXA)221 b FL(has)g(tw)-11 b(o)220 b(ar)-20 b(guments:)317 b(a)221 b(quadratic)h(form)f Fz(q)260 b FL(and)222 b(a)f(v)-17 b(ector)222 b(whose)g(components)-913 22554 y(are)277 b(the)g(form')-61 b(s)276 b(v)-28 b(ariables.)-913 23882 y FJ(QXA)277 b FL(returns)g(the)g(matrix)g Fz(A)g FL(associated)h(with) f Fz(q)40 b FL(.)-913 25211 y(T)-89 b(yping)279 b(:)9583 26613 y FA(QXA)p Fw(\()p FA(2)247 b FI(\003)g FA(S1)f FI(\003)g FA(S2)277 b Fz(;)462 b Fw([)p FA(S1)p Fz(;)184 b FA(S2)p Fw(]\))-913 28679 y FL(gi)-28 b(v)-17 b(es)278 b(:)12799 30080 y Fw([[)p FA(0)p Fz(;)184 b FA(1)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(0)p Fw(]])-913 32146 y FL(and)278 b(with)f FJ(SHOW)13377 32398 y Fp(\024)14515 33283 y Fw(0)1108 b(1)14515 34612 y(1)g(0)17283 32398 y Fp(\025)-913 36739 y SDict begin H.S end -913 36739 a -913 36739 a SDict begin 12 H.A end -913 36739 a -913 36739 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.8.4) cvn H.B /DEST pdfmark end -913 36739 a 1051 x FE(4.8.4)1329 b(Sylv)-13 b(ester)331 b(decomposition)g(:)412 b Fq(SYLVESTER)-913 39912 y FJ(SYLVESTER)211 b FL(has)g(one)h(ar)-20 b(gument:)312 b(a)211 b(symmetric)f(matrix)h(representing)h(a)f(quadratic)-913 41240 y(form)276 b Fz(q)40 b FL(.)-913 42569 y FJ(SYLVESTER)351 b FL(returns)e(a)h(list)f(of)g(tw)-11 b(o)350 b(elements:)490 b(the)350 b(diagonal)i(elements)f(of)f(the)-913 43897 y(diagonal)271 b(matrix)e Fz(B)325 b FL(\(obtained)271 b(e)-17 b(xpressing)271 b Fz(q)309 b FL(as)269 b(a)h(sum)f(of)g (squares\))h(and)g(the)g(ma-)-913 45225 y(trix)276 b(of)h(change)i(of)e (base)h Fz(Q)p FL(.)-913 46554 y(W)-89 b(e)277 b(ha)-22 b(v)-17 b(e:)12152 47498 y Ft(t)12542 47955 y Fz(Q)246 b FI(\001)g Fz(B)301 b FI(\001)247 b Fz(Q)308 b Fw(=)f Fz(A)-913 50021 y FL(T)-89 b(yping:)9754 51423 y FA(SYLVESTER)p Fw(\([[)p FA(0)p Fz(;)184 b FA(1)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(0)p Fw(]]\))-913 53489 y FL(gi)-28 b(v)-17 b(es)278 b(:)10104 55522 y FI(f)p Fw([)11097 54773 y FA(1)p 11097 55267 582 45 v 11097 56281 a(2)11812 55522 y Fz(;)184 b FI(\000)p FA(2)p Fw(])278 b Fz(;)15099 53961 y Fp(\024)16659 54844 y Fw(1)1607 b(1)16369 55742 y Fs(\000)p Fv(1)p 16369 55923 1134 45 v 16715 56559 a(2)18875 55742 y(1)p 18875 55923 442 45 v 18875 56559 a(2)20003 53961 y Fp(\025)20587 55522 y FI(g)p Black Black eop end end %%Page: 54 54 TeXDict begin HPSdict begin 54 53 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.54) cvn H.B /DEST pdfmark end -913 1926 a Black FL(54)p FD(CHAPTER)213 b(4.)532 b(THE)211 b FJ(CAS)g FD(FUNCTIONS)h(ONL)-111 b(Y)212 b(V)-149 b(ALID)210 b(IN)g(HOME)h(SCREEN)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (section.4.9) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FF(4.9)1593 b(Functions)399 b(of)f(multiple)h(v)-16 b(ariables)-913 5619 y SDict begin H.S end -913 5619 a -913 5619 a SDict begin 12 H.A end -913 5619 a -913 5619 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.9.1) cvn H.B /DEST pdfmark end -913 5619 a 1216 x FE(4.9.1)1329 b(The)331 b(r)-24 b(otor)333 b(:)412 b Fq(CURL)-913 8911 y FL(Here)277 b Fz(n)308 b Fw(=)g(3)p FL(.)-913 10239 y FJ(CURL)477 b FL(has)h(tw)-11 b(o)477 b(ar)-20 b(guments:)745 b(the)478 b(v)-28 b(alue)479 b([)p Fz(E)64 b Fw(1)p Fz(;)184 b(E)64 b Fw(2)p Fz(;)184 b(E)64 b Fw(3)p FL(])480 b(of)d(a)g(v)-17 b(ectorial)479 b(func-)-913 11567 y(tion)317 b(\(application)h(from)f Fz(R)10049 11166 y Fv(3)10861 11567 y FL(in)g Fz(R)12889 11166 y Fv(3)13384 11567 y FL(\))f(and)i(a)f(v)-17 b(ector)319 b(of)d Fz(R)21947 11166 y Fv(3)22760 11567 y FL(denoting)i(the)g(v)-28 b(ariable)-913 12896 y(names.)-913 14224 y FJ(CURL)277 b FL(returns)g(the)g(rotor)g(of)g([)p Fz(E)64 b Fw(1)p Fz(;)184 b(E)64 b Fw(2)p Fz(;)184 b(E)64 b Fw(3)p FL(].)-913 16852 y FA(CURL)p Fw(\([)p FA(E1)p Fz(;)184 b FA(E2)p Fz(;)g FA(E3)p Fw(])p Fz(;)g Fw([)p FA(S1)p Fz(;)g FA(S2)p Fz(;)g FA(S3)p Fw(]\))317 b(=)308 b([)14852 16103 y Fz(@)61 b FA(E3)p 14852 16597 1812 45 v 14852 17611 a Fz(@)g FA(S2)17042 16852 y FI(\000)18282 16103 y Fz(@)g FA(E2)p 18282 16597 V 18282 17611 a Fz(@)g FA(S3)20227 16852 y Fz(;)20851 16103 y(@)g FA(E1)p 20851 16597 V 20851 17611 a Fz(@)g FA(S3)23042 16852 y FI(\000)24281 16103 y Fz(@)g FA(E3)p 24281 16597 V 24281 17611 a Fz(@)g FA(S1)26226 16852 y Fz(;)26851 16103 y(@)g FA(E2)p 26851 16597 V 26851 17611 a Fz(@)g FA(S1)29041 16852 y FI(\000)30281 16103 y Fz(@)g FA(E1)p 30281 16597 V 30281 17611 a Fz(@)g FA(S2)32225 16852 y Fw(])-913 19226 y FL(T)-89 b(ype:)5012 20555 y FA(CURL)p Fw(\([)p FA(S1)248 b FI(\003)f FA(S3)p Fz(;)184 b FI(\000)p FA(S2)13959 20097 y Fo(2)14427 20555 y Fz(;)g FA(2)247 b FI(\003)g FA(S1)f FI(\003)h FA(S2)p Fw(])p Fz(;)184 b Fw([)p FA(S1)p Fz(;)g FA(S2)p Fz(;)g FA(S3)p Fw(]\))-913 22353 y FL(Y)-122 b(ou)278 b(obtain:)10486 23682 y Fw([)p FA(2)246 b FI(\003)h FA(S1)p Fz(;)184 b FA(S1)247 b FI(\000)f FA(S2)g FI(\003)h FA(2)p Fz(;)184 b FA(0)p Fw(])-913 25519 y SDict begin H.S end -913 25519 a -913 25519 a SDict begin 12 H.A end -913 25519 a -913 25519 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.9.2) cvn H.B /DEST pdfmark end -913 25519 a 1273 x FE(4.9.2)1329 b(The)331 b(gradient)h(:)412 b Fq(DERIV)-913 28868 y FJ(DERIV)328 b FL(has)f(tw)-11 b(o)328 b(ar)-20 b(guments:)445 b(the)327 b(v)-28 b(alue)330 b(of)d(an)h(application)h Fz(E)64 b Fw(1)328 b FL(from)f Fz(R)29192 28466 y Ft(n)30120 28868 y FL(in)g Fz(R)-913 30196 y FL(and)278 b(a)f(v)-17 b(ector)278 b(of)f Fz(R)6805 29794 y Ft(n)7684 30196 y FL(denoting)i(the)e(v)-28 b(ariable)279 b(names.)-913 31524 y FJ(DERIV)e FL(returns)g(the)g(gradient)i(of)d Fz(E)64 b Fw(1)278 b FL(\()p Fw([)15288 31089 y Ft(@)48 b(E)43 b Fv(1)p 15288 31270 1655 45 v 15341 31906 a Ft(@)48 b(S)42 b Fv(1)17074 31524 y Fz(;)17699 31089 y Ft(@)48 b(E)43 b Fv(1)p 17699 31270 V 17752 31906 a Ft(@)48 b(S)42 b Fv(2)19486 31524 y Fz(;)20111 31089 y Ft(@)48 b(E)43 b Fv(1)p 20111 31270 V 20164 31906 a Ft(@)48 b(S)42 b Fv(3)21898 31524 y Fw(])277 b FL(if)f Fz(n)308 b Fw(=)g(3)p FL(\).)-913 32853 y(T)-89 b(ype:)5623 34181 y FA(DERIV)p Fw(\()p FA(2)247 b FI(\003)g FA(S1)11748 33724 y Fo(2)12461 34181 y FI(\003)g FA(S2)f FI(\000)g FA(S1)g FI(\003)h FA(S3)19146 33724 y Fo(3)19613 34181 y Fz(;)184 b Fw([)p FA(S1)p Fz(;)g FA(S2)p Fz(;)g FA(S3)p Fw(]\))-913 35980 y FL(After)276 b(simpli\002cation,)i(you)g(obtain:)6227 38092 y Fw([)p FA(4)247 b FI(\001)f FA(S2)h FI(\001)f FA(S1)h FI(\000)e FA(S3)13554 37635 y Fo(3)14022 38092 y Fz(;)184 b FA(2)247 b FI(\001)f FA(S1)17056 37635 y Fo(2)17524 38092 y Fz(;)184 b FI(\000)p Fw(\()p FA(3)247 b FI(\001)g FA(S1)f FI(\001)g FA(S3)23811 37635 y Fo(2)24279 38092 y Fw(\)])-913 39930 y SDict begin H.S end -913 39930 a -913 39930 a SDict begin 12 H.A end -913 39930 a -913 39930 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.9.3) cvn H.B /DEST pdfmark end -913 39930 a 1273 x FE(4.9.3)1329 b(The)331 b(di)-13 b(v)g(er)g(gence)330 b(:)412 b Fq(DIV)-913 43278 y FJ(DIV)225 b FL(has)g(tw)-11 b(o)225 b(ar)-20 b(guments:)319 b(the)225 b(v)-28 b(alue)227 b([)p Fz(E)64 b Fw(1)p Fz(;)184 b(E)64 b Fw(2)p Fz(;)184 b(E)64 b Fw(3)p FL(])227 b(of)e(a)h(v)-17 b(ectorial)226 b(function)g(\(ap-)-913 44606 y(plication)278 b(from)e Fz(R)6516 44205 y Ft(n)7395 44606 y FL(in)h Fz(R)9383 44205 y Ft(n)9984 44606 y FL(\))g(and)h(a)f (v)-17 b(ector)278 b(of)f Fz(R)18348 44205 y Ft(n)19227 44606 y FL(denoting)i(the)e(v)-28 b(ariable)279 b(names.)-913 45935 y FJ(DIV)e FL(returns)g(the)g(di)-28 b(v)-17 b(er)d(gence)281 b(of)c([)p Fz(E)64 b Fw(1)p Fz(;)184 b(E)64 b Fw(2)p Fz(;)184 b(E)64 b Fw(3)p FL(].)806 48588 y FA(DIV)p Fw(\([)p FA(E1)p Fz(;)184 b FA(E2)p Fz(;)g FA(E3)p Fw(])p Fz(;)g Fw([)p FA(S1)p Fz(;)g FA(S2)p Fz(;)g FA(S3)p Fw(]\))317 b(=)15682 47839 y Fz(@)61 b FA(E1)p 15682 48333 1812 45 v 15682 49347 a Fz(@)g FA(S1)17873 48588 y Fw(+)19113 47839 y Fz(@)g FA(E2)p 19113 48333 V 19113 49347 a Fz(@)g FA(S2)21303 48588 y Fw(+)22543 47839 y Fz(@)g FA(E3)p 22543 48333 V 22543 49347 a Fz(@)g FA(S3)24764 48588 y FL(\(here)277 b Fz(n)308 b Fw(=)g(3\))-913 52028 y FL(T)-89 b(yping:)5303 53357 y FA(DIV)p Fw(\([)p FA(S1)248 b FI(\003)e FA(S3)p Fz(;)184 b FI(\000)p FA(S2)13668 52900 y Fo(2)14137 53357 y Fz(;)g FA(2)247 b FI(\003)f FA(S1)h FI(\003)f FA(S2)p Fw(])p Fz(;)184 b Fw([)p FA(S1)p Fz(;)g FA(S2)p Fz(;)g FA(S3)p Fw(]\))-913 55156 y FL(gi)-28 b(v)-17 b(es:)12970 56484 y FA(S3)246 b FI(\000)g FA(2)g FI(\003)h FA(S2)p Black Black eop end end %%Page: 55 55 TeXDict begin HPSdict begin 55 54 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.55) cvn H.B /DEST pdfmark end -913 1926 a Black FD(4.10.)620 b(EQ)-11 b(U)-44 b(A)-123 b(TIONS)22978 b FL(55)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.9.4) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(4.9.4)1329 b(The)331 b(hessian)g(:)412 b Fq(HESS)-913 6232 y FJ(HESS)240 b FL(has)f(tw)-11 b(o)240 b(ar)-20 b(guments:)326 b(the)240 b(v)-28 b(alue)241 b(of)e(an)h(application)i Fz(E)64 b Fw(1)239 b FL(from)h Fz(R)27530 5830 y Ft(n)28370 6232 y FL(in)g Fz(R)247 b FL(and)-913 7560 y(a)277 b(v)-17 b(ector)278 b(of)f Fz(R)4930 7159 y Ft(n)5808 7560 y FL(denoting)j(the)d(v)-28 b(ariable)278 b(names.)-913 8889 y FJ(HESS)356 b FL(returns)f(a)h(list)f(containing)j(the)e (hessian)g(of)g Fz(E)64 b Fw(1)p FL(,)375 b(the)356 b(gradient)h(of)f Fz(E)64 b Fw(1)356 b FL(and)-913 10217 y(the)277 b(v)-17 b(ector)278 b(of)f(the)h(v)-28 b(ariable)278 b(names.)-913 11545 y(T)-89 b(yping)279 b(:)5870 12900 y FA(HESS)p Fw(\()p FA(2)248 b FI(\003)e FA(S1)11414 12443 y Fo(2)12127 12900 y FI(\003)h FA(S2)f FI(\000)g FA(S1)g FI(\003)h FA(S3)277 b Fz(;)462 b Fw([)p FA(S1)p Fz(;)184 b FA(S2)p Fz(;)g FA(S3)p Fw(]\))-913 14919 y FL(gi)-28 b(v)-17 b(es)278 b(:)-913 18664 y FI(f)-360 16439 y Fp(2)-360 18432 y(4)932 17325 y FA(2)246 b FI(\001)g FA(2)g FI(\001)h FA(S2)1107 b(2)246 b FI(\001)h FA(2)f FI(\001)g FA(S1)1108 b FI(\000)p Fw(1)932 18653 y FA(2)246 b FI(\001)g FA(2)g FI(\001)h FA(S1)2792 b Fw(0)3223 b(0)2186 19982 y FI(\000)p Fw(1)4047 b(0)3223 b(0)12960 16439 y Fp(3)12960 18432 y(5)14160 18664 y Fz(;)461 b Fw([)p FA(4)247 b FI(\001)f FA(S1)h FI(\001)f FA(S2)h FI(\000)e FA(S3)p Fz(;)184 b FA(2)248 b FI(\001)e FA(S1)25290 18207 y Fo(2)25758 18664 y Fz(;)184 b FI(\000)p FA(S1)p Fw(])278 b Fz(;)462 b Fw([)p FA(S1)p Fz(;)184 b FA(S2)p Fz(;)g FA(S3)p Fw(])p FI(g)-913 22372 y FL(No)-28 b(w)-72 b(,)277 b(to)g(obtain)h(the)g (critical)f(points)g(of)g Fz(E)64 b Fw(1)p FL(,)277 b(you)h(must)e (type:)4585 24825 y FA(SOLVE)p Fw(\([)p FA(4)p Fz(:)p FA(S1)p Fz(:)p FA(S2)249 b FI(\000)d FA(S3)p Fz(;)184 b FA(2)p Fz(:)p FA(S1)16805 24368 y Fo(2)17275 24825 y Fz(;)g FI(\000)p FA(S1)p Fw(])278 b Fz(;)462 b Fw([)p FA(S1)p Fz(;)184 b FA(S2)p Fz(;)g FA(S3)p Fw(]\))-913 26760 y SDict begin H.S end -913 26760 a -913 26760 a SDict begin 12 H.A end -913 26760 a -913 26760 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.9.5) cvn H.B /DEST pdfmark end -913 26760 a 1282 x FE(4.9.5)1329 b(The)331 b(laplacian)h(:)413 b Fq(LAPL)-913 30134 y FJ(LAPL)240 b FL(has)f(tw)-11 b(o)240 b(ar)-20 b(guments:)326 b(the)240 b(v)-28 b(alue)241 b(of)e(an)h(application)i Fz(E)64 b Fw(1)239 b FL(from)h Fz(R)27530 29732 y Ft(n)28370 30134 y FL(in)g Fz(R)247 b FL(and)-913 31462 y(a)277 b(v)-17 b(ector)278 b(of)f Fz(R)4930 31061 y Ft(n)5808 31462 y FL(denoting)j(the)d(v)-28 b(ariable)278 b(names.)-913 32940 y FJ(LAPL)f FL(returns)g(the)g (laplacian)i(of)e Fz(E)64 b Fw(1)277 b FL(\()14685 32505 y Ft(@)15206 32171 y Fm(2)15637 32505 y Ft(E)43 b Fv(1)p 14684 32686 2087 45 v 14737 33322 a Ft(@)48 b(S)42 b Fv(1)16285 33101 y Fm(2)17149 32940 y Fw(+)18389 32505 y Ft(@)18910 32171 y Fm(2)19342 32505 y Ft(E)h Fv(1)p 18389 32686 V 18442 33322 a Ft(@)48 b(S)42 b Fv(2)19990 33101 y Fm(2)20854 32940 y Fw(+)22094 32505 y Ft(@)22615 32171 y Fm(2)23046 32505 y Ft(E)h Fv(1)p 22094 32686 V 22147 33322 a Ft(@)48 b(S)42 b Fv(3)23695 33101 y Fm(2)24590 32940 y FL(if)276 b Fz(n)308 b Fw(=)f(3)p FL(\).)-913 34269 y(T)-89 b(ype:)5637 35624 y FA(LAPL)p Fw(\()p FA(2)247 b FI(\003)g FA(S1)11181 35166 y Fo(2)11894 35624 y FI(\003)f FA(S2)h FI(\000)e FA(S1)i FI(\003)f FA(S3)18578 35166 y Fo(3)19322 35624 y Fz(;)462 b Fw([)p FA(S1)p Fz(;)184 b FA(S2)p Fz(;)g FA(S3)p Fw(]\))-913 37642 y FL(Y)-122 b(ou)278 b(obtain:)11053 38997 y FA(4)246 b FI(\003)g FA(S2)h FI(\000)f FA(6)g FI(\003)g FA(S1)h FI(\003)f FA(S3)-913 40872 y SDict begin H.S end -913 40872 a -913 40872 a SDict begin 12 H.A end -913 40872 a -913 40872 a SDict begin [ /View [/XYZ H.V] /Dest (section.4.10) cvn H.B /DEST pdfmark end -913 40872 a 1910 x FF(4.10)1593 b(Equations)-913 44268 y SDict begin H.S end -913 44268 a -913 44268 a SDict begin 12 H.A end -913 44268 a -913 44268 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.10.1) cvn H.B /DEST pdfmark end -913 44268 a 1226 x FE(4.10.1)1329 b(Members)330 b(of)j(an)e (equation)h(:)412 b Fq(EXLR)-913 47586 y FJ(EXLR)277 b FL(has)g(an)h(equation)h(as)e(ar)-20 b(gument)279 b(\(you)f(ha)-22 b(v)-17 b(e)278 b(to)f(type)h FJ(==)p FL(\).)-913 48914 y FJ(EXLR)d FL(returns)g(a)h(list)d(containing)278 b(the)d(left)g(and)h (right)f(hand)i(sides)e(of)g(the)g(equation.)-913 50243 y(T)-89 b(ype:)8753 52926 y FA(EXLR)p Fw(\()p FA(XQ)p Fw(\()p FA(2)248 b FI(\001)e FA(S1)h Fw(+)f FA(1)307 b Fw(==)h FA(3)p Fz(=)p FA(2)p Fw(\)\))-913 54945 y FL(Y)-122 b(ou)278 b(obtain:)11726 56299 y FI(f)p FA(2)247 b FI(\001)f FA(S1)h Fw(+)f FA(1)p Fz(;)184 b FA(3)p Fz(=)p FA(2)p FI(g)p Black Black eop end end %%Page: 56 56 TeXDict begin HPSdict begin 56 55 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.56) cvn H.B /DEST pdfmark end -913 1926 a Black FL(56)p FD(CHAPTER)213 b(4.)532 b(THE)211 b FJ(CAS)g FD(FUNCTIONS)h(ONL)-111 b(Y)212 b(V)-149 b(ALID)210 b(IN)g(HOME)h(SCREEN)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (section.4.11) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FF(4.11)1593 b(Matrices)-913 5410 y SDict begin H.S end -913 5410 a -913 5410 a SDict begin 12 H.A end -913 5410 a -913 5410 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.1) cvn H.B /DEST pdfmark end -913 5410 a 1647 x FE(4.11.1)1329 b(Matrix)333 b(to)f(list)h(:)412 b Fq(AXL)-913 9354 y FL(When)258 b FJ(AXL)g FL(is)f(gi)-28 b(v)-17 b(en)259 b(a)f(matrix)g(as)f(ar)-20 b(gument,)263 b(it)256 b(returns)i(the)g (same)g(matrix)f(re)-28 b(writ-)-913 10682 y(ten)277 b(as)g(a)g(list)f(of)g(lists.)342 b(V)-66 b(ice)277 b(v)-17 b(ersa,)277 b FJ(AXL)h FL(transforms)e(a)h(list)f(of)g(lists)g(into)h (a)g(matrix.)-913 12011 y(T)-89 b(yping:)9918 13689 y FA(AXL)p Fw(\()p 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FI(g)p Fz(;)g FI(f)p FA(3)p Fz(;)g FA(4)p FI(gg)p Fw(\))-913 40157 y FL(gi)-28 b(v)-17 b(es:)12509 41836 y Fw([[)p FA(1)p Fz(;)184 b FA(0)p Fz(:)p FA(5)p Fz(;)g Fw([)p FA(3)p Fz(;)g FA(4)p Fw(]])-913 44256 y SDict begin H.S end -913 44256 a -913 44256 a SDict begin 12 H.A end -913 44256 a -913 44256 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.2) cvn H.B /DEST pdfmark end -913 44256 a 1390 x FE(4.11.2)1329 b(Symbolic)331 b(matrix)i(to)f(numeric)f(matrix)h (:)412 b Fq(AXM)-913 47943 y FL(If)422 b(gi)-28 b(v)-17 b(en)425 b(a)f(symbolic)g(matrix)f(as)g(ar)-20 b(gument,)461 b FJ(AXM)423 b FL(returns)g(an)h(equi)-28 b(v)g(alent)426 b(\(b)-22 b(ut)-913 49272 y(approximate\))279 b(numeric)f(matrix.)-913 50600 y(T)-89 b(ype:)10930 52278 y FA(AXM)p Fw(\([[)p FA(1)p Fz(=)p FA(2)p Fz(;)184 b FA(2)p Fw(])p Fz(;)g Fw([)p FA(3)p Fz(;)g FA(4)p Fw(]]\))-913 54621 y FL(Y)-122 b(ou)278 b(obtain:)12355 56299 y Fw([[)p FA(0)p Fz(:)p FA(5)p Fz(;)184 b FA(2)p Fw(])p Fz(;)g Fw([)p FA(3)p Fz(;)g FA(4)p Fw(]])p Black Black eop end end %%Page: 57 57 TeXDict begin HPSdict begin 57 56 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.57) cvn H.B /DEST pdfmark end -913 1926 a Black FD(4.11.)620 b(MA)-123 b(TRICES)23659 b FL(57)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.3) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(4.11.3)1329 b(Basis)332 b(v)-13 b(ectors)331 b(:)413 b Fq(BASIS)-913 6215 y FJ(BASIS)339 b FL(has)f(as)g(ar)-20 b(gument)340 b(a)f(list)e(of)h(v)-17 b(ectors,)354 b(de\002ning)341 b(a)e(v)-17 b(ectorial)339 b(subspace)h(of)-913 7544 y Fz(R)-63 7142 y Ft(n)538 7544 y FL(.)-913 8872 y FJ(BASIS)305 b FL(returns)g(a)g(list)e(containing)k(the)e(v)-17 b(ectors)306 b(of)f(a)g(basis)f(of)h(the)g(v)-17 b(ectorial)306 b(sub-)-913 10200 y(space.)-913 11529 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b(type:)7680 13844 y FA(BASIS)p Fw(\()p FI(f)p Fw([)p FA(1)p Fz(;)184 b FA(2)p Fz(;)g FA(3)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(1)p Fz(;)g FA(1)p Fw(])p Fz(;)g Fw([)p FA(2)p Fz(;)g FA(3)p Fz(;)g FA(4)p Fw(])p FI(g)p Fw(\))-913 16159 y FL(Y)-122 b(ou)278 b(obtain:)10911 17487 y FI(f)p Fw([)p FA(1)p Fz(;)184 b FA(0)p Fz(;)g FI(\000)p FA(1)p Fw(])p Fz(;)466 b Fw([)p FA(0)p Fz(;)184 b FA(1)p Fz(;)g FA(2)p Fw(])p FI(g)-913 19361 y SDict begin H.S end -913 19361 a -913 19361 a SDict begin 12 H.A end -913 19361 a -913 19361 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.4) cvn H.B /DEST pdfmark end -913 19361 a 1273 x FE(4.11.4)1329 b(Cholesk)-20 b(y)331 b(matrix)i(:)412 b Fq(CHOLESKY)-913 22710 y FJ(CHOLESKY)278 b FL(has)f(as)g(ar)-20 b(gument)279 b(a)e(square)h(matrix)f FJ(M)p FL(,)f(positi)-28 b(v)-17 b(e)279 b(by)e(de\002nition.)-913 24038 y FJ(CHOLESKY)h FL(returns)f(an)g(upper)h(triangular)g(matrix)f FJ(P)g FL(so)g(that:)-913 24965 y Fo(t)-446 25366 y FA(P)246 b FI(\003)g FA(P)308 b Fw(=)f FA(M)-913 26695 y FL(T)-89 b(yping:)10044 28023 y FA(CHOLESKY)p Fw(\([[)p FA(1)p Fz(;)184 b FA(1)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(5)p Fw(]]\))-913 29943 y FL(gi)-28 b(v)-17 b(es)278 b(:)13377 30339 y Fp(\024)14515 31225 y Fw(1)1108 b(1)14515 32553 y(0)g(2)17283 30339 y Fp(\025)-913 34549 y SDict begin H.S end -913 34549 a -913 34549 a SDict begin 12 H.A end -913 34549 a -913 34549 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.5) cvn H.B /DEST pdfmark end -913 34549 a 1026 x FE(4.11.5)1329 b(Map)252 b(an)g(operator)g(to)h(a)g (diagonalizable)f(matrix)h(:)372 b Fq(DIAGMAP)-913 37650 y FJ(DIAGMAP)342 b FL(has)f(tw)-11 b(o)341 b(ar)-20 b(guments:)473 b(a)342 b(diagonalizable)i(matrix)d(and)i(an)f(e)-17 b(xpression)-913 38979 y(de\002ning)279 b(an)f(holomorphic)h(operator) -61 b(.)-913 40307 y FJ(DIAGMAP)278 b FL(applies)f(the)h(operator)g(to) f(the)g(matrix,)g(and)h(returns)f(the)g(result.)-913 41635 y(T)-89 b(yping:)8205 42964 y FA(DIAGMAP)p Fw(\([[)p FA(1)p Fz(;)184 b FA(1)p Fw(])p Fz(;)g Fw([)p FA(0)p Fz(;)g FA(2)p Fw(]])p Fz(;)g FA(EXP)p Fw(\()p FA(S1)p Fw(\)\))-913 44884 y FL(gi)-28 b(v)-17 b(es:)8518 45280 y Fp(\024)9655 46165 y FA(EXP)q Fw(\(1\))1108 b FA(EXP)q Fw(\(2\))247 b FI(\000)f FA(EXP)p Fw(\(1\))10957 47494 y(0)4666 b FA(EXP)p Fw(\(2\))22142 45280 y Fp(\025)-913 49490 y SDict begin H.S end -913 49490 a -913 49490 a SDict begin 12 H.A end -913 49490 a -913 49490 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.6) cvn H.B /DEST pdfmark end -913 49490 a 1026 x FE(4.11.6)1329 b(Hadamard)331 b(pr)-24 b(oduct)331 b(:)412 b Fq(HADAMARD)-913 52591 y FJ(HADAMARD)278 b FL(has)f(tw)-11 b(o)277 b(matrices)g Fz(A)g FL(and)h Fz(B)56 b FL(,)276 b(with)h(the)g(same)h(size,)f(as)f(ar)-20 b(guments.)-913 53919 y FJ(HADAMARD)278 b FL(returns)f(the)g (element-by-element)j(product)f(between)g Fz(A)e FL(and)h Fz(B)56 b FL(.)-913 55248 y(T)-89 b(ype:)4385 56576 y FA(HADAMARD)p Fw(\()p FA(XQ)p Fw(\([[)p FA(1)p Fz(=)p FA(2)p Fz(;)184 b FA(2)p Fw(])p Fz(;)g Fw([)p FA(3)p Fz(;)g FA(4)p Fw(]]\))p Fz(;)g FA(XQ)p Fw(\([[)p FA(5)p Fz(;)g FA(6)p Fw(])p Fz(;)g Fw([)p FA(7)p Fz(;)g FA(8)p Fw(]]\)\))p Black Black eop end end %%Page: 58 58 TeXDict begin HPSdict begin 58 57 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R 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y(Therefore,)271 b Fz(Q)p Fw(\()p Fz(x)p Fw(\))f FL(is)e(also)h(the)g(co-matrix)h(of)e Fz(A)217 b FI(\000)f Fz(x)g FI(\001)g Fz(I)356 b FL(and)270 b(the)f(follo)-28 b(wing)270 b(holds:)-913 39665 y Fz(Q)p Fw(\()p Fz(x)p Fw(\))561 b(=)g Fz(I)245 b FI(\001)159 b Fz(x)5269 39263 y Ft(n)p Fs(\000)p Fv(1)7164 39665 y Fw(+)g Fz(:::)g Fw(+)g Fz(B)11124 39831 y Fv(0)11621 39665 y FL(,)258 b(where)d Fz(B)15954 39831 y Fv(0)16703 39665 y FL(is)e(the)g(co-matrix)i(of)e Fz(A)g FL(\(with)g(the)h(sign) -913 40993 y(e)-17 b(xchanged)281 b(if)276 b Fz(n)h FL(is)f(e)-28 b(v)-17 b(en!\).)-913 42321 y(T)-89 b(ype:)10930 43650 y FA(MAD)p Fw(\([[)p FA(1)p Fz(;)184 b FA(1)p Fz(=)p FA(2)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(3)p Fw(]]\))-913 45445 y FL(or)10192 46773 y FA(MAD)p Fw(\()p FI(ff)p FA(1)p Fz(;)g FA(1)p Fz(=)p FA(2)p FI(g)p Fz(;)g FI(f)p FA(1)p Fz(;)g FA(3)p FI(gg)p Fw(\))-913 48568 y FL(Y)-122 b(ou)278 b(obtain:)-913 50674 y FI(f)p FA(5)p Fz(=)p FA(2)p Fz(;)184 b Fw([[)p FA(6)p Fz(=)p FA(5)p 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14156 15292 a(5)11481 15808 y(4)p 11481 15989 V 11481 16626 a(5)13162 16244 y FI(\000)14156 15808 y Fv(3)p 14156 15989 V 14156 16626 a(5)15284 14025 y Fp(\025)16052 15586 y Fz(;)16543 14025 y Fp(\024)17682 14911 y Fw(5)1107 b(7)17682 16239 y(0)g(1)20449 14025 y Fp(\025)21034 15586 y FI(g)-913 18304 y SDict begin H.S end -913 18304 a -913 18304 a SDict begin 12 H.A end -913 18304 a -913 18304 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.17) cvn H.B /DEST pdfmark end -913 18304 a 1035 x FE(4.11.17)1329 b(System)331 b(to)i(matrix)f(:)412 b Fq(SYST2MAT)-913 21433 y FJ(SYST2MAT)457 b FL(has)g(tw)-11 b(o)457 b(ar)-20 b(guments:)704 b(a)456 b(v)-17 b(ector)458 b(containing)h(a)e(system)g(of)f(linear)-913 22761 y(equations)239 b(\()p FJ(=)e FL(or)h FJ(==)g FL(is)e(not)i(allo)-28 b(wed)240 b(and)f(so)e FJ(=0)h FL(is)f(omitted\),)245 b(and)239 b(a)f(v)-17 b(ector)239 b(whose)-913 24090 y(elements)278 b(are)f(the)g(system')-61 b(s)276 b(v)-28 b(ariables.)-913 25418 y FJ(SYST2MAT)278 b 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clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.63) cvn H.B /DEST pdfmark end -913 1926 a Black FD(4.12.)620 b(THE)278 b(PERMUT)-103 b(A)-123 b(TIONS)18383 b FL(63)p Black -913 4140 a(gi)-28 b(v)-17 b(es:)12355 5535 y Fw([[)p FA(1)p Fz(;)184 b FA(3)p Fw(])p Fz(;)g Fw([)p FA(0)p Fz(:)p FA(5)p Fz(;)g FA(4)p Fw(]])-913 7532 y SDict begin H.S end -913 7532 a -913 7532 a SDict begin 12 H.A end -913 7532 a -913 7532 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.11.19) cvn H.B /DEST pdfmark end -913 7532 a 1295 x FE(4.11.19)1329 b(V)-122 b(andermonde)329 b(matrix)k(:)412 b Fq(VANDERMONDE)-913 10946 y FJ(VANDERMONDE)327 b FL(has)f(as)g(ar)-20 b(gument)328 b(a)e(v)-17 b(ector)327 b(whose)g(components)i(are)d(denoted)-913 12274 y(by)277 b Fz(x)1103 12440 y Ft(i)1748 12274 y FL(for)g Fz(i)308 b Fw(=)f(0)p Fz(::n)248 b FI(\000)e Fw(1)p FL(.)-913 13602 y FJ(VANDERMONDE)345 b FL(returns)f(the)g(corresponding)i(V)-123 b(andermonde)348 b(matrix)c(\(the)g Fz(k)35 b FL(-th)-913 14931 y(ro)-28 b(w)304 b(of)f(the)h(matrix)f(is)g(the)h(v)-17 b(ector)305 b(whose)g(components)h(are)e Fz(x)23982 14529 y Ft(k)23982 15219 y(i)24829 14931 y FL(for)f Fz(i)357 b Fw(=)g(0)p Fz(::n)267 b FI(\000)f Fw(1)-913 16259 y FL(and)278 b Fz(k)342 b Fw(=)308 b(0)p Fz(::n)248 b FI(\000)e Fw(1)p FL(\).)-913 17587 y(T)-89 b(ype:)10324 18983 y FA(VANDERMONDE)p Fw(\([)p FA(1)p Fz(;)184 b FA(2)p Fz(;)g FA(3)p Fw(]\))-913 21043 y FL(Y)-122 b(ou)278 b(obtain:)9809 22438 y Fw([[)p FA(1)p Fz(;)184 b FA(1)p Fz(;)g FA(1)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(2)p Fz(;)g FA(4)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(3)p Fz(;)g FA(9)p Fw(]])-913 24498 y FL(Suppose)279 b(that)e FJ(E1=6)p FL(,)g FJ(E2=2*S1+S1*S2)i FL(and)f(then)g(type:)9452 26978 y FA(VANDERMONDE)p Fw(\([)p FA(E1)p Fz(;)184 b FA(E2)p Fz(;)g FA(E3)p Fw(]\))-913 29458 y FL(Y)-122 b(ou)278 b(obtain)g(pressing)f FJ(ENTER)p FL(,)g(then)h FI(")p FL(,)f(and)i(then)e FJ(SHOW)p FL(:)5249 31006 y Fp(2)5249 32998 y(4)6554 31892 y Fw(1)4355 b(6)7961 b(36)6541 33220 y FA(1)1107 b(2)246 b FI(\001)g FA(S1)h Fw(+)f FA(S1)g FI(\001)g FA(S2)1108 b Fw(\()p FA(2)246 b FI(\001)h FA(S1)f Fw(+)g FA(S1)g FI(\001)h FA(S2)p Fw(\))24236 32818 y Fo(2)6541 34548 y FA(1)4036 b(E3)7395 b(E3)20877 34147 y Fo(2)25257 31006 y Fp(3)25257 32998 y(5)-913 36791 y SDict begin H.S end -913 36791 a -913 36791 a SDict begin 12 H.A end -913 36791 a -913 36791 a SDict begin [ /View [/XYZ H.V] /Dest (section.4.12) cvn H.B /DEST pdfmark end -913 36791 a 1232 x FF(4.12)1593 b(The)399 b(permutations)-913 40539 y FL(Permutations)278 b(are)f(de\002ned)j (with)d(the)g(image)h(list)e FI(f)p Fz(P)154 b Fw(\(1\))p Fz(;)184 b(P)154 b Fw(\(2\))p Fz(:::P)g Fw(\()p Fz(n)p Fw(\))p FI(g)p FL(.)-913 41868 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b(de\002nining)f(the)f(permutation)g Fz(P)430 b FL(as)277 b Fz(P)461 b Fw(=)307 b FI(f)p Fw(3)p Fz(;)184 b Fw(2)p Fz(;)g Fw(1)p FI(g)282 b FL(means)c(that:)-913 43196 y Fz(P)154 b Fw(\(1\))308 b(=)f(3)p Fz(;)463 b(P)154 b Fw(\(2\))308 b(=)f(2)p Fz(;)463 b(P)154 b Fw(\(3\))308 b(=)f(1)p FL(.)-913 44524 y(A)294 b(c)-17 b(ycle)295 b(is)e(denoted)k(with)d(a)g(list)e(containing)297 b(the)d(images)h(of)f (an)g(element)i(through)-913 45853 y(the)277 b(c)-17 b(ycle;)279 b(for)d(e)-17 b(xample,)279 b(de\002ning)h(a)d(c)-17 b(ycle)279 b Fz(C)356 b FL(as)277 b Fz(C)386 b Fw(=)308 b FI(f)p Fw(3)p Fz(;)184 b Fw(2)p Fz(;)g Fw(1)p FI(g)281 b FL(means)d(that:)-913 47181 y Fz(C)79 b Fw(\(3\))309 b(=)e(2)p Fz(;)463 b(C)79 b Fw(\(2\))309 b(=)e(1)p Fz(;)463 b(C)79 b Fw(\(1\))309 b(=)e(3)p FL(.)-913 48509 y(Accordingly)-72 b(,)279 b(a)e(decomposition)j(into)d(c)-17 b(ycles)278 b(is)e(denoted)k(with)c(a)i(list)d(of)i(lists.)-913 51557 y SDict begin H.S end -913 51557 a -913 51557 a SDict begin 12 H.A end -913 51557 a -913 51557 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.12.1) cvn H.B /DEST pdfmark end -913 51557 a 1573 x FE(4.12.1)1329 b(Cycles)332 b(to)g(permutation)f(:) 413 b Fq(C2P)-913 55248 y FJ(C2P)277 b FL(has)g(a)g(list)f(of)h(c)-17 b(ycles)278 b(as)f(ar)-20 b(gument.)-913 56576 y FJ(C2P)417 b FL(returns)f(the)h(permutation)i(ha)-22 b(ving)418 b(the)f(input)h(list)d(as)i(decomposition)i(into)p Black Black eop end end %%Page: 64 64 TeXDict begin HPSdict begin 64 63 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.64) cvn H.B /DEST pdfmark end -913 1926 a Black FL(64)p FD(CHAPTER)213 b(4.)532 b(THE)211 b FJ(CAS)g FD(FUNCTIONS)h(ONL)-111 b(Y)212 b(V)-149 b(ALID)210 b(IN)g(HOME)h(SCREEN)p Black -913 4140 a FL(c)-17 b(ycles)278 b(\(see)f(also)g FJ(P2C)p FL(\).)-913 5468 y(T)-89 b(yping:)10223 6796 y FA(C2P)p Fw(\()p FI(ff)p FA(1)p Fz(;)184 b FA(3)p Fz(;)g FA(5)p FI(g)p Fz(;)g FI(f)p FA(2)p Fz(;)g FA(4)p FI(gg)p Fw(\))-913 8766 y FL(gi)-28 b(v)-17 b(es:)12632 10095 y FI(f)p FA(3)p Fz(;)184 b FA(4)p Fz(;)g FA(5)p Fz(;)g FA(2)p Fz(;)g FA(1)p FI(g)-913 11983 y SDict begin H.S end -913 11983 a -913 11983 a SDict begin 12 H.A end -913 11983 a -913 11983 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.12.2) cvn H.B /DEST pdfmark end -913 11983 a 1273 x FE(4.12.2)1329 b(Composition)332 b(:)412 b Fq(CIRC)-913 15332 y FJ(CIRC)277 b FL(has)g(tw)-11 b(o)277 b(permutations)i(as)e(ar) -20 b(guments.)-913 16660 y FJ(CIRC)421 b FL(returns)f(the)i (permutation)g(obtained)h(by)e(composition:)633 b(\()p Fw(1)26297 16259 y FL(st)27090 16660 y(ar)-20 b(gument)355 b FI(\016)-913 18164 y Fw(2)-360 17762 y FL(nd)802 18164 y(ar)-20 b(gument)r(\).)-913 19492 y(T)-89 b(yping:)7803 20821 y FA(CIRC)p Fw(\()p FI(f)p FA(3)p Fz(;)184 b FA(4)p Fz(;)g FA(5)p Fz(;)g FA(2)p Fz(;)g FA(1)p FI(g)p Fz(;)g FI(f)p FA(2)p Fz(;)g FA(1)p Fz(;)g FA(4)p Fz(;)g FA(3)p Fz(;)g FA(5)p FI(g)p Fw(\))-913 22790 y FL(gi)-28 b(v)-17 b(es)278 b(:)12632 24119 y FI(f)p FA(4)p Fz(;)184 b FA(3)p Fz(;)g FA(2)p Fz(;)g FA(5)p Fz(;)g FA(1)p FI(g)-913 26008 y SDict begin H.S end -913 26008 a -913 26008 a SDict begin 12 H.A end -913 26008 a -913 26008 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.12.3) cvn H.B /DEST pdfmark end -913 26008 a 1273 x FE(4.12.3)1329 b(Decomposition)331 b(into)i(cycles)p Fq(P2C)-913 29356 y FJ(P2C)277 b FL(has)g(a)g (permutation)i(as)e(ar)-20 b(gument.)-913 30685 y FJ(P2C)277 b FL(returns)g(its)e(decomposition)280 b(into)d(c)-17 b(ycles)279 b(and)f(its)d(signature.)-913 32013 y(T)-89 b(yping:)11330 33341 y FA(P2C)p Fw(\()p FI(f)p FA(3)p Fz(;)184 b FA(4)p Fz(;)g FA(5)p Fz(;)g FA(2)p Fz(;)g FA(1)p FI(g)p Fw(\))-913 35311 y FL(gi)-28 b(v)-17 b(es)278 b(:)10004 36639 y FI(fff)p FA(1)p Fz(;)184 b FA(3)p Fz(;)g FA(5)p FI(g)p Fz(;)g FI(f)p FA(2)p Fz(;)g FA(4)p FI(gg)p Fz(;)g FI(\000)p FA(1)p FI(g)-913 38653 y SDict begin H.S end -913 38653 a -913 38653 a SDict begin 12 H.A end -913 38653 a -913 38653 a SDict begin [ /View [/XYZ H.V] /Dest (section.4.13) cvn H.B /DEST pdfmark end -913 38653 a 1716 x FF(4.13)1593 b(The)399 b(isometries)-913 41533 y SDict begin H.S end -913 41533 a -913 41533 a SDict begin 12 H.A end -913 41533 a -913 41533 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.13.1) cvn H.B /DEST pdfmark end -913 41533 a 1531 x FE(4.13.1)1329 b(Isometry')-49 b(s)331 b(characteristics)g(elements)g(:)412 b Fq(ISOM)-913 45140 y FJ(ISOM)395 b FL(has)g(as)g(ar)-20 b(gument)396 b(a)f(matrix)g (representing)h(a)g(tw)-11 b(o)395 b(or)f(three-dimensional)-913 46468 y(linear)277 b(isometry)-72 b(.)-913 47796 y FJ(ISOM)336 b FL(returns)g(the)h(list)e(of)h(the)h(isometry')-61 b(s)336 b(characteristics)g(elements)h(and)h(either)-913 49125 y FJ(+1)277 b FL(\(for)f(direct)h(isometries\))f(or)h FJ(-1)g FL(\(for)f(indirect)h(isometries\).)-913 50453 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b(type:)8216 52850 y FA(ISOM)p Fw(\([[)p FA(0)p Fz(;)184 b FA(0)p Fz(;)g FA(1)p Fw(])p Fz(;)g Fw([)p FA(0)p Fz(;)g FA(1)p Fz(;)g FA(0)p Fw(])p Fz(;)g Fw([)p FA(1)p Fz(;)g FA(0)p Fz(;)g FA(0)p Fw(]]\))-913 55248 y FL(Y)-122 b(ou)278 b(obtain:)12000 56576 y FI(f)p Fw([)p FA(1)p Fz(;)184 b FA(0)p Fz(;)g FI(\000)p FA(1)p Fw(])p Fz(;)g FI(\000)p FA(1)p FI(g)p Black Black eop end end %%Page: 65 65 TeXDict begin HPSdict begin 65 64 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.65) cvn H.B /DEST pdfmark end -913 1926 a Black FD(4.13.)620 b(THE)278 b(ISOMETRIES)20185 b FL(65)p Black -913 4140 a(This)292 b(result)g(means)i(that)f(the)g (isometry)f(is)g(a)h(simmetry)f(with)h(respect)g(to)g(the)g(plane)-913 5468 y Fz(x)522 b FI(\000)h Fz(z)632 b Fw(=)585 b(0)p FL(.)-913 6796 y(If)276 b(you)i(type:)10070 9013 y FA(ISOM)p Fw(\()12957 7367 y FI(p)p 13880 7367 582 45 v 13880 8264 a FA(2)p 12957 8759 1504 45 v 13419 9773 a(2)14840 9013 y FI(\001)15394 7453 y Fp(\024)16531 8338 y Fw(1)1108 b FI(\000)p Fw(1)16531 9666 y(1)1969 b(1)20160 7453 y Fp(\025)20744 9013 y Fw(\))-913 11721 y FL(Y)-122 b(ou)278 b(obtain:)11313 13180 y FI(f)p FA(ACOS)p Fw(\()14620 12180 y Fp(p)p 15728 12180 431 45 v 15728 13180 a Fw(\()q FA(1)p Fz(=)p FA(2)p Fw(\))p Fz(;)184 b FA(1)p FI(g)-913 15595 y FL(therefore,)277 b(this)f(isometry)h(is)f(a)i(rotation)f(of) 16498 14846 y Fz(\031)p 16498 15340 671 45 v 16557 16354 a Fw(4)17578 15595 y FL(radians.)-913 18169 y SDict begin H.S end -913 18169 a -913 18169 a SDict begin 12 H.A end -913 18169 a -913 18169 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.13.2) cvn H.B /DEST pdfmark end -913 18169 a 1070 x FE(4.13.2)1329 b(Isometry')-49 b(s)331 b(matrix)h(:)413 b Fq(MKISOM)-913 21397 y FL(In)f(a)h(three-dimensional)h(space,)447 b FJ(MKISOM)413 b FL(has)g(the)g(list)e(of)h(the)h(characteristics)-913 22725 y(elements)324 b(of)g(an)g(isometry)-72 b(,)336 b(and)325 b(either)f FJ(+1)g FL(\(denoting)h(a)f(direct)g(isometry\))g (or)f FJ(-1)-913 24054 y FL(\(denoting)278 b(an)g(indirect)f (isometry\))g(as)g(ar)-20 b(guments,)-913 25382 y(In)329 b(a)h(tw)-11 b(o-dimensional)331 b(space,)344 b FJ(MKISOM)330 b FL(has)g(the)g(characteristic)g(element)h(of)f(an)-913 26711 y(isometry)239 b(\(either)g(an)i(angle)f(or)g(a)f(v)-17 b(ector\))241 b(and)f(either)g FJ(+1)f FL(\(denoting)j(a)d(direct)h (isom-)-913 28039 y(etry\))276 b(or)h FJ(-1)g FL(\(denoting)i(an)f (indirect)f(isometry\))g(as)g(ar)-20 b(guments.)-913 29367 y FJ(MKISOM)277 b FL(returns)g(the)h(matrix)f(representing)h(the) f(gi)-28 b(v)-17 b(en)279 b(isometry)-72 b(.)-913 30696 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b(type:)9245 33218 y FA(MKISOM)p Fw(\()p FI(f)p Fw([)p FI(\000)p FA(1)p Fz(;)184 b FA(2)p Fz(;)g FI(\000)p FA(1)p Fw(])p Fz(;)g(\031)40 b FI(g)p Fz(;)184 b FA(1)p Fw(\))-913 35741 y FL(Y)-122 b(ou)278 b(simplify)e(the)i(answer)f(typing)h(:)-913 37069 y FJ(XQ\()f FL(then)h FI(")f FL(and)h FJ(COPY)p FL(,)f(then)h FJ(\))665 b(ENTER)-913 38397 y FL(this)276 b(gi)-28 b(v)-17 b(es)278 b(the)g(matrix)f(of)f(a)i(rotation)f(with)g (axis)g Fw([)p FI(\000)p Fw(1)p Fz(;)184 b Fw(2)p Fz(;)g FI(\000)p Fw(1])281 b FL(and)d(angle)h Fz(\031)316 b FL(:)11125 40031 y Fp(2)11125 42024 y(4)12550 40478 y Fs(\000)p Fv(2)p 12550 40659 1134 45 v 12896 41296 a(3)15056 40478 y Fs(\000)p Fv(2)p 15056 40659 V 15401 41296 a(3)18253 40478 y(1)p 18253 40659 442 45 v 18253 41296 a(3)12550 41812 y Fs(\000)p Fv(2)p 12550 41993 1134 45 v 12896 42629 a(3)15747 41812 y(1)p 15747 41993 442 45 v 15747 42629 a(3)17561 41812 y Fs(\000)p Fv(2)p 17561 41993 1134 45 v 17907 42629 a(3)13242 43146 y(1)p 13242 43327 442 45 v 13242 43963 a(3)15056 43146 y Fs(\000)p Fv(2)p 15056 43327 1134 45 v 15401 43963 a(3)17561 43146 y Fs(\000)p Fv(2)p 17561 43327 V 17907 43963 a(3)19381 40031 y Fp(3)19381 42024 y(5)-913 46035 y FL(F)-17 b(or)277 b(e)-17 b(xample,)280 b(type:)11592 47494 y FA(MKISOM)p Fw(\()p FI(f)p Fz(\031)40 b FI(g)p Fz(;)184 b FI(\000)p FA(1)p Fw(\))-913 49618 y FL(Y)-122 b(ou)278 b(obtain)g(the)f(matrix)g(of)g(a)g(symmetry)h (with)f(respect)g(to)g(the)h(origin:)8518 52140 y Fw([[)p FI(\000)p FA(1)p Fz(;)184 b FA(0)p Fz(;)g FA(0)p Fw(])p Fz(;)g Fw([)p FA(0)p Fz(;)g FI(\000)p FA(1)p Fz(;)g FA(0)p Fw(])p Fz(;)g Fw([)p FA(0)p Fz(;)g FA(0)p Fz(;)g FI(\000)p FA(1)p Fw(]])-913 54663 y FL(T)-89 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1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.81) cvn H.B /DEST pdfmark end -913 1926 a Black FD(5.3.)620 b(MODULAR)278 b(CALCULA)-123 b(TIONS)15452 b FL(81)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.3.7) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(5.3.7)1329 b(In)-53 b(v)-13 b(erse)330 b(in)i(Z/pZ)g(:)g Fq(INVMOD)-913 6275 y FJ(INVMOD)277 b FL(has)h(as)f(an)g(ar)-20 b(gument)279 b(an)f(inte)-17 b(ger)-61 b(.)-913 7604 y FJ(INVMOD)277 b FL(calculates)i(the)e(in)-44 b(v)-17 b(erse)278 b(of)f(the)g(inte)-17 b(ger)278 b(in)f Fz(Z)18 b(=pZ)79 b FL(.)-913 8932 y(T)-89 b(yping:)13158 10355 y FA(INVMOD)p Fw(\()p FA(5)p Fw(\))-913 12442 y FL(gi)-28 b(v)-17 b(es)278 b(\(since)f Fw(5)246 b FI(\002)g(\000)p Fw(5)308 b(=)g FI(\000)p Fw(25)g(=)g(1)277 b(\()61 b(mo)31 b(d)340 b(13\))p FL(\))278 b(:)14901 14940 y FI(\000)p FA(5)-913 16793 y SDict begin H.S end -913 16793 a 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1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.88) cvn H.B /DEST pdfmark end -913 1926 a Black FL(88)661 b FD(CHAPTER)279 b(5.)620 b FJ(CAS)277 b FD(REFERENCE)j(FR)-44 b(OM)278 b(THE)f(EQ)-11 b(U)-44 b(A)-123 b(TION)278 b(EDIT)-20 b(OR)p Black -913 4140 a FL(produces:)13606 5931 y FA(ATAN)p Fw(\()16493 5183 y FA(4)p 16493 5677 582 45 v 16493 6691 a(3)17208 5931 y Fw(\))-913 8332 y SDict begin H.S end -913 8332 a -913 8332 a SDict begin 12 H.A end -913 8332 a -913 8332 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.6.2) cvn H.B /DEST pdfmark end -913 8332 a 1031 x FE(5.6.2)1329 b(Conjugate)332 b(of)h(a)f(complex)f(number)f(:)412 b Fq(CONJ)-913 11448 y FL(T)-89 b(yping:)12081 12791 y FA(CONJ)p Fw(\()p FA(1)248 b Fw(+)e FA(2)g FI(\001)g FA(i)p Fw(\))-913 14798 y FL(produces)278 b(the)g(result:)13674 16142 y FA(1)246 b FI(\000)g FA(2)g FI(\001)h FA(i)-885 18149 y FL(N)56 b FB(O)21 b(T)56 b(E)28 b 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31025 y FA(CONJ)p Fw(\()p FA(1)247 b Fw(+)f FA(i)p Fw(\))-913 33032 y FL(whereas:)9323 34375 y FA(SUBST)p Fw(\()p FA(CONJ)p Fw(\()p FA(Z)p Fw(\))p Fz(;)184 b FA(Z)312 b Fw(=)307 b FA(1)246 b Fw(+)g FA(i)p Fw(\))-913 36383 y FL(produces:)14364 37726 y FA(1)h Fw(+)f FA(i)-913 39733 y FL(Of)300 b(course,)308 b(if)300 b(you)i(ha)-22 b(v)-17 b(e)303 b(selected)f FJ(Cmplx)665 b(vars)p FL(,)307 b(and)302 b FJ(Z)f FL(is)f(not)h(in)g(the)h(list)d (that)-913 41062 y(contains)278 b(the)f(v)-28 b(ariable)279 b FJ(REALASSUME)p FL(,)f(then)g(typing:)9323 43507 y FA(SUBST)p Fw(\()p FA(CONJ)p Fw(\()p FA(Z)p Fw(\))p Fz(;)184 b FA(Z)312 b Fw(=)307 b FA(1)246 b Fw(+)g FA(i)p Fw(\))-913 45952 y FL(gi)-28 b(v)-17 b(es:)12772 47295 y FA(CONJ)p Fw(\()p FA(1)247 b Fw(+)f FA(i)p Fw(\))-913 49213 y SDict begin H.S end -913 49213 a -913 49213 a SDict begin 12 H.A end -913 49213 a -913 49213 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.6.3) cvn H.B /DEST pdfmark end -913 49213 a 1278 x FE(5.6.3)1329 b(Equation)331 b(of)i(the)f(line)g(:)412 b Fq(DROITE)-913 52576 y FJ(DROITE)277 b FL(tak)-11 b(es)278 b(tw)-11 b(o)277 b(comple)-17 b(x)279 b(numbers)g Fz(z)16492 52742 y Fv(1)16988 52576 y Fz(;)462 b(z)18272 52742 y Fv(2)19045 52576 y FL(as)277 b(parameters.)-913 53905 y FJ(DROITE)211 b FL(returns)f(the)h(equation)i(of)d(the)h(line)g (through)h(the)f(Cartesian)g(points,)224 b Fz(z)30048 54071 y Fv(1)30545 53905 y Fz(;)395 b(z)31762 54071 y Fv(2)32258 53905 y FL(.)-913 55233 y(T)-89 b(yping:)10687 56576 y FA(DROITE)p Fw(\(\()p FA(1)p Fz(;)184 b FA(2)p Fw(\))p Fz(;)g Fw(\()p FA(0)p Fz(;)g FA(1)p Fw(\)\))p Black Black eop end end %%Page: 89 89 TeXDict begin HPSdict begin 89 88 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.89) cvn H.B /DEST pdfmark end -913 1926 a Black FD(5.7.)620 b(ALGEBRAIC)279 b(EXPRESSIONS)15757 b FL(89)p Black -913 4140 a(or:)10964 5803 y FA(DROITE)p Fw(\()p FA(1)247 b Fw(+)f FA(2)g FI(\001)h FA(i)p Fz(;)184 b FA(i)p Fw(\))-913 8131 y FL(returns:)12369 9795 y FA(Y)307 b Fw(=)h FA(X)246 b FI(\000)g FA(1)g Fw(+)g FA(2)-913 12123 y FL(Then,)278 b FJ(ENTER)g FL(produces:)13336 13786 y FA(Y)308 b Fw(=)f FA(X)246 b Fw(+)g FA(1)-913 16125 y SDict begin H.S end -913 16125 a -913 16125 a SDict begin 12 H.A end -913 16125 a -913 16125 a SDict begin [ /View [/XYZ H.V] /Dest (section.5.7) cvn H.B /DEST pdfmark end -913 16125 a 2012 x FF(5.7)1593 b(Algebraic)398 b(Expr)-29 b(essions)-913 20824 y FL(All)323 b(functions)h(in)f(this)g(section)h (can)h(be)f(found)h(in)f(the)g FJ(ALGB)g FL(menu)h(on)f(the)g(menu)-913 22152 y(bar)-61 b(.)-913 24289 y SDict begin H.S end -913 24289 a -913 24289 a SDict begin 12 H.A end -913 24289 a -913 24289 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.7.1) cvn H.B /DEST pdfmark end -913 24289 a 1646 x FE(5.7.1)1329 b(F)-33 b(actors)331 b(o)-13 b(v)g(er)332 b(the)f(integers)h(:)412 b Fq(COLLECT)-913 28223 y FJ(COLLECT)278 b FL(has)f(an)h(e)-17 b(xpression)278 b(as)f(a)g(parameter)-61 b(.)-913 29552 y FJ(COLLECT)379 b FL(combines)h(lik)-11 b(e)379 b(terms,)403 b(and)379 b(f)-11 b(actors)379 b(the)f(e)-17 b(xpression)380 b(o)-17 b(v)g(er)381 b(the)d(inte-)-913 30880 y(gers.)-913 32208 y(Examples)278 b(:)-913 33537 y(T)-89 b(o)278 b(f)-11 b(actor)277 b(o)-17 b(v)g(er)278 b(the)g(inte)-17 b(gers:)14104 35200 y Fz(x)14737 34743 y Fv(2)15479 35200 y FI(\000)246 b Fw(4)-913 37528 y FL(type:)11666 39192 y FA(COLLECT)p Fw(\()p FA(X)16744 38735 y Fo(2)17459 39192 y FI(\000)g FA(4)p Fw(\))-913 41520 y FL(which)278 b(gi)-28 b(v)-17 b(es)278 b(in)f(real)g(mode)h(:)11846 44178 y Fw(\()p FA(X)247 b Fw(+)f FA(2)p Fw(\))g FI(\001)h Fw(\()p FA(X)f FI(\000)g FA(2)p Fw(\))-913 46837 y FL(T)-89 b(o)278 b(f)-11 b(actor)277 b(o)-17 b(v)g(er)278 b(the)g(inte)-17 b(gers:)14104 48501 y Fz(x)14737 48044 y Fv(2)15479 48501 y FI(\000)246 b Fw(2)-913 50829 y FL(type:)11666 52492 y FA(COLLECT)p Fw(\()p FA(X)16744 52035 y Fo(2)17459 52492 y FI(\000)g FA(2)p Fw(\))-913 54820 y FL(which)278 b(gi)-28 b(v)-17 b(es:)14131 56484 y FA(X)14712 56027 y Fo(2)15425 56484 y FI(\000)246 b FA(2)p Black Black eop end end %%Page: 90 90 TeXDict begin HPSdict begin 90 89 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest 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Fw(\))p -780 39426 41819 45 v 19547 40440 a FA(16)-913 42043 y SDict begin H.S end -913 42043 a -913 42043 a SDict begin 12 H.A end -913 42043 a -913 42043 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.7.4) cvn H.B /DEST pdfmark end -913 42043 a 1026 x FE(5.7.4)1329 b(Change)331 b(the)h(order)f(of)i(v)-13 b(ariables)331 b(:)412 b Fq(REORDER)-913 45145 y FJ(REORDER)380 b FL(has)g(tw)-11 b(o)380 b(ar)-20 b(guments:)551 b(an)380 b(e)-17 b(xpression)381 b(and)g(the)g(names)f(of)g(v)-28 b(ariables)-913 46473 y(separated)278 b(with)f FJ(AND)p FL(.)-913 47802 y FJ(REORDER)400 b FL(reorders)g(the)f(input)i(e)-17 b(xpression)401 b(follo)-28 b(wing)400 b(the)g(order)g(of)f(v)-28 b(ariables)-913 49130 y(gi)g(v)-17 b(en)279 b(by)e(its)f(second)j(ar) -20 b(gument.)-913 50458 y(T)-89 b(yping:)2813 52807 y FA(REORDER)p Fw(\()p FA(X)7891 52350 y Fo(2)8605 52807 y Fw(+)246 b FA(2)h FI(\001)f FA(X)g FI(\001)h FA(A)f Fw(+)g FA(A)14989 52350 y Fo(2)15702 52807 y Fw(+)g FA(Z)17390 52350 y Fo(2)18103 52807 y FI(\000)g FA(X)g FI(\001)g FA(Z)p Fz(;)184 b FA(A)278 b(AND)g(X)f(AND)g(Z)p Fw(\))-913 55156 y FL(gi)-28 b(v)-17 b(es:)8692 56484 y FA(A)9273 56027 y Fo(2)9986 56484 y Fw(+)246 b FA(2)g FI(\001)h FA(X)f FI(\001)g FA(A)g Fw(+)g FA(X)16369 56027 y Fo(2)17082 56484 y FI(\000)g FA(Z)h FI(\001)f FA(X)g Fw(+)g FA(Z)22085 56027 y Fo(2)p Black Black eop end end %%Page: 91 91 TeXDict begin HPSdict begin 91 90 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.91) cvn H.B /DEST pdfmark end -913 1926 a Black FD(5.8.)620 b(POL)-111 b(YNOMIALS)21865 b FL(91)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.7.5) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(5.7.5)1329 b(P)-13 b(artial)332 b(simpli\002cations)f(:)412 b Fq(SEVAL)-913 6356 y FJ(SEVAL)277 b FL(has)h(an)f(e)-17 b(xpression)279 b(as)e(ar)-20 b(gument.)-913 7684 y FJ(SEVAL)237 b FL(simpli\002es)g (the)h(e)-17 b(xpression,)246 b(operating)239 b(on)f(all)f(b)-22 b(ut)238 b(the)f(top-le)-28 b(v)-17 b(el)239 b(operator)-913 9012 y(of)276 b(the)i(e)-17 b(xpression.)-913 10341 y(T)-89 b(yping)279 b(:)7252 11891 y FA(SEVAL)p Fw(\()p FA(SIN)p Fw(\()p FA(3)249 b FI(\001)d FA(X)g FI(\000)g FA(X)p Fw(\))h(+)f FA(SIN)p Fw(\()p FA(X)h Fw(+)f FA(X)p Fw(\)\))-913 14105 y FL(gi)-28 b(v)-17 b(es)278 b(:)10379 15655 y FA(SIN)p Fw(\()p FA(2)247 b FI(\001)g FA(X)p Fw(\))g(+)f FA(SIN)p Fw(\()p FA(2)h FI(\001)f FA(X)p Fw(\))-913 17883 y SDict begin H.S end -913 17883 a -913 17883 a SDict begin 12 H.A end -913 17883 a -913 17883 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.7.6) cvn H.B /DEST pdfmark end -913 17883 a 1347 x FE(5.7.6)1329 b(Substitute)331 b(a)h(v)-13 b(alue)332 b(f)-33 b(or)332 b(a)g(v)-13 b(ariable)332 b(:)412 b Fq(|)-913 21446 y FJ(|)296 b FL(is)f(an)i(in\002x)h(operator) f(used)g(to)g(substitute)f(a)g(v)-28 b(alue)299 b(for)c(a)i(v)-28 b(ariable)298 b(in)e(an)h(e)-17 b(xpres-)-913 22774 y(sion)277 b(\(a)f(bit)h(lik)-11 b(e)277 b(the)h(follo)-28 b(wing)278 b(function)g FJ(SUBST)p FL(\).)-913 24102 y FJ(|)423 b FL(has)h(tw)-11 b(o)424 b(parameters:)637 b(an)424 b(e)-17 b(xpression)425 b(dependent)i(on)d(a)g(parameter)-44 b(,)461 b(and)425 b(an)-913 25431 y(equality)278 b (\(parameter=substitute)g(v)-28 b(alue\).)-913 26759 y FJ(|)277 b FL(substitutes)f(the)h(speci\002ed)i(v)-28 b(alue)279 b(for)d(the)i(v)-28 b(ariable)278 b(in)f(the)h(e)-17 b(xpression.)-913 28087 y(T)-89 b(yping:)12492 29637 y FA(X)13073 29180 y Fo(2)13786 29637 y FI(\000)246 b FA(1)277 b FI(j)16058 29932 y FA(X)308 b Fw(=)g FA(2)-913 31851 y FL(gi)-28 b(v)-17 b(es:)14131 33401 y FA(2)14712 32944 y Fo(2)15425 33401 y FI(\000)246 b FA(1)-913 35445 y SDict begin H.S end -913 35445 a -913 35445 a SDict begin 12 H.A end -913 35445 a -913 35445 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.7.7) cvn H.B /DEST pdfmark end -913 35445 a 1531 x FE(5.7.7)1329 b(Substitute)331 b(a)h(v)-13 b(alue)332 b(f)-33 b(or)332 b(a)g(v)-13 b(ariable)332 b(:)412 b Fq(SUBST)-913 39192 y FJ(SUBST)318 b FL(has)g(tw)-11 b(o)318 b(parameters:)426 b(an)319 b(e)-17 b(xpression)319 b(dependent)i(on)e(a)f(parameter)-44 b(,)329 b(and)-913 40520 y(an)277 b(equality)i(\(parameter=substitute)f(v)-28 b(alue\).)-913 41849 y FJ(SUBST)277 b FL(substitutes)g(the)g (speci\002ed)i(v)-28 b(alue)279 b(for)d(the)i(v)-28 b(ariable)278 b(in)f(the)g(e)-17 b(xpression.)-913 43177 y(T)-89 b(yping:)10682 44727 y FA(SUBST)p Fw(\()p FA(A)14598 44270 y Fo(2)15313 44727 y Fw(+)246 b FA(1)p Fz(;)184 b FA(A)309 b Fw(=)e FA(2)p Fw(\))-913 46941 y FL(gi)-28 b(v)-17 b(es:)14131 48491 y FA(2)14712 48034 y Fo(2)15425 48491 y Fw(+)246 b FA(1)-913 50659 y SDict begin H.S end -913 50659 a -913 50659 a SDict begin 12 H.A end -913 50659 a -913 50659 a SDict begin [ /View [/XYZ H.V] /Dest (section.5.8) cvn H.B /DEST pdfmark end -913 50659 a 1974 x FF(5.8)1593 b(P)-32 b(olynomials)-913 55248 y FL(All)327 b(functions)h(\(e)-17 b(xcept)330 b(DEGREE\))f(in)e(this)g(section)i(can)g(be)f(found)h(in)f (the)g FJ(MATH)-913 56576 y FL(k)-11 b(e)-17 b(y')-61 b(s)277 b(Polynom.)346 b(menu.)p Black Black eop end end %%Page: 92 92 TeXDict begin HPSdict begin 92 91 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.92) cvn H.B /DEST pdfmark end -913 1926 a Black FL(92)661 b FD(CHAPTER)279 b(5.)620 b FJ(CAS)277 b FD(REFERENCE)j(FR)-44 b(OM)278 b(THE)f(EQ)-11 b(U)-44 b(A)-123 b(TION)278 b(EDIT)-20 b(OR)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.1) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(5.8.1)1329 b(Resolution)252 b(of)h Fq (A[X]U[X]+B[X]V[X]=C[X])f FE(:)h Fq(ABCUV)-913 6215 y FL(This)360 b(command)j(applies)e(the)g(B\351zout)h(identity)e(lik)-11 b(e)361 b FJ(EGCD)g FL(b)-22 b(ut,)381 b(no)-28 b(w)-72 b(,)382 b(the)361 b(ar)-20 b(gu-)-913 7544 y(ments)210 b(are)h(three)g(polynomials,)226 b FJ(A,)664 b(B,)h(C)210 b FL(\()p FJ(C)g FL(must)h(be)g(a)g(multiple)f(of)h FJ(GCD\(A,B\))p FL(\):)-913 8872 y FJ(ABCUV\(A[X],B[X],C[X]\))j FL(returns)c FJ(U[X])665 b(AND)f(V[X])p FL(,)211 b(where)h FJ(U,V)f FL(sat-)-913 10200 y(isfy)276 b(the)h(follo)-28 b(wing:)8719 12262 y FA(C)p Fw([)p FA(X)p Fw(])309 b(=)e FA(U)p Fw([)p FA(X)p Fw(])247 b FI(\003)g FA(A)p Fw([)p FA(X)p Fw(])g(+)f FA(V)p Fw([)p FA(X)p Fw(])h FI(\003)g FA(B)p Fw([)p FA(X)p Fw(])-913 14323 y FL(T)-89 b(yping:)7350 15651 y FA(ABCUV)p Fw(\()p FA(X)11266 15194 y Fo(2)11980 15651 y Fw(+)246 b FA(2)g FI(\001)g FA(X)h Fw(+)f FA(1)p Fz(;)184 b FA(X)18055 15194 y Fo(2)18769 15651 y FI(\000)246 b FA(1)p Fz(;)184 b FA(X)247 b Fw(+)f FA(1)p Fw(\))-913 17420 y FL(gi)-28 b(v)-17 b(es:)13329 18447 y FA(1)p 13329 18942 582 45 v 13329 19956 a(2)14320 19196 y(AND)16473 18447 y FI(\000)p FA(1)p 16473 18942 1443 45 v 16903 19956 a(2)-913 21257 y FL(T)-89 b(yping:)7116 22585 y FA(ABCUV)p Fw(\()p FA(X)11032 22128 y Fo(2)11746 22585 y Fw(+)246 b FA(2)g FI(\001)h FA(X)f Fw(+)g FA(1)p Fz(;)184 b FA(X)17821 22128 y Fo(2)18535 22585 y FI(\000)246 b FA(1)p Fz(;)184 b FA(X)21295 22128 y Fo(3)22010 22585 y Fw(+)245 b FA(1)p Fw(\))-913 24353 y FL(gi)-28 b(v)-17 b(es:)8748 25580 y FA(X)9329 25178 y Fo(2)10042 25580 y FI(\000)246 b FA(X)g Fw(+)g FA(1)p 8748 26074 4917 45 v 10916 27088 a(2)14074 26329 y(AND)523 b FI(\000)17580 25580 y FA(X)18161 25178 y Fo(2)18874 25580 y FI(\000)246 b FA(X)g Fw(+)g FA(1)p 17580 26074 V 19747 27088 a(2)-913 28640 y SDict begin H.S end -913 28640 a -913 28640 a SDict begin 12 H.A end -913 28640 a -913 28640 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.2) cvn H.B /DEST pdfmark end -913 28640 a 1026 x FE(5.8.2)1329 b(Chinese)330 b(r)-24 b(emainders)331 b(:)412 b Fq(CHINREM)-913 31741 y FJ(CHINREM)320 b FL(has)g(tw)-11 b(o)320 b FJ(AND)g FL(as)f(ar)-20 b(guments)321 b(:)428 b(each)322 b FJ(AND)e FL(has)f(tw)-11 b(o)320 b(polynomials)h(as)-913 33069 y(components.)-913 34398 y FJ(CHINREM\(A[X])666 b(AND)f(R[X],B[X])g (AND)g(Q[X]\))271 b FL(returns)f(an)i FJ(AND)e FL(with)-913 35726 y(tw)-11 b(o)265 b(polynomials)j(as)d(components)j FJ(P[X])e FL(and)h FJ(S[X])p FL(.)f(This)g(polyonimals)h FJ(P[X])-913 37055 y FL(and)249 b FJ(S[X])g FL(satisfying)f(the)h (follo)-28 b(wing)249 b(relations)f(when)i FJ(GCD\(R[X],Q[X]\)=1)p FL(:)-913 38383 y FA(S)p Fw([)p FA(X)p Fw(])308 b(=)g FA(R)p Fw([)p FA(X)p Fw(])247 b FI(\001)g FA(Q)p Fw([)p FA(X)p Fw(])q FL(,)-913 39711 y FA(P)p Fw([)p FA(X)p Fw(])308 b(=)g FA(A)p Fw([)p FA(X)p Fw(]\()p FA(modR)p Fw([)p FA(X)p Fw(]\))281 b FL(and)d FA(P)p Fw([)p FA(X)p Fw(])309 b(=)e FA(B)p Fw([)p FA(X)p Fw(]\()p FA(modQ)p Fw([)p FA(X)p Fw(]\))t FL(.)-913 41040 y(There)296 b(is)f(al)-11 b(w)g(ays)295 b(a)h(solution)g FJ(P[X])p FL(,)f(if)g FJ(R[X])g FL(and)i FJ(Q[X])f FL(are)f(mutually)i(primes,)-913 42368 y(and)278 b(all)e(solutions)i(are)f(congruent)i(modulo)g FA(S)p Fw([)p FA(X)p Fw(])309 b(=)e FA(R)p Fw([)p FA(X)p Fw(])247 b FI(\001)g FA(Q)p Fw([)p FA(X)p Fw(])q FL(.)-913 43696 y(Find)278 b(the)f(solutions)g Fz(P)154 b Fw([)p Fz(X)87 b Fw(])277 b FL(of:)7219 44861 y Fp(\032)8603 45747 y Fz(P)154 b Fw([)p Fz(X)87 b Fw(])308 b(=)1107 b Fz(X)364 b Fw(\()61 b(mo)31 b(d)340 b Fz(X)18599 45345 y Fv(2)19341 45747 y Fw(+)246 b(1\))8603 47075 y Fz(P)154 b Fw([)p Fz(X)87 b Fw(])308 b(=)1107 b Fz(X)333 b FI(\000)246 b Fw(1)277 b(\()61 b(mo)31 b(d)340 b Fz(X)20505 46673 y Fv(2)21247 47075 y FI(\000)246 b Fw(1\))-913 49068 y FL(T)-89 b(yping:)6084 50396 y FA(CHINREM)p Fw(\()p FA(X)279 b(AND)e(X)14042 49939 y Fo(2)14755 50396 y Fw(+)246 b FA(1)p Fz(;)184 b FA(X)247 b FI(\000)f FA(1)277 b(AND)g(X)22328 49939 y Fo(2)23041 50396 y FI(\000)246 b FA(1)p Fw(\))-913 52164 y FL(gi)-28 b(v)-17 b(es:)8461 54140 y FI(\000)9455 53391 y FA(X)10036 52989 y Fo(2)10749 53391 y FI(\000)246 b FA(2)g FI(\001)h FA(X)f Fw(+)g FA(1)p 9455 53885 6297 45 v 12313 54899 a(2)16162 54140 y(AND)523 b FI(\000)19668 53391 y FA(X)20249 52989 y Fo(4)20962 53391 y FI(\000)246 b FA(1)p 19668 53885 2983 45 v 20868 54899 a(2)-913 56576 y FL(that)277 b(is,)f Fz(P)154 b Fw([)p Fz(X)87 b Fw(])307 b(=)h FI(\000)7269 56140 y Ft(X)8055 55806 y Fm(2)8487 56140 y Fs(\000)p Fv(2)p Fs(\001)p Ft(X)60 b Fv(+1)p 7269 56321 4522 45 v 9309 56958 a(2)12199 56576 y Fw(\()h(mo)31 b(d)310 b FI(\000)16360 56140 y Ft(X)17146 55806 y Fm(4)17579 56140 y Fs(\000)p Fv(1)p 16360 56321 2352 45 v 17315 56958 a(2)18845 56576 y Fw(\))p Black Black eop end end %%Page: 93 93 TeXDict begin HPSdict begin 93 92 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.93) cvn H.B /DEST pdfmark end -913 1926 a Black FD(5.8.)620 b(POL)-111 b(YNOMIALS)21865 b FL(93)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.3) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(5.8.3)1329 b(Cyclotomic)332 b(polynomial)h(:)412 b Fq(CYCLOTOMIC)-913 6263 y FJ(CYCLOTOMIC)278 b FL(has)f(an)h(inte)-17 b(ger)278 b Fz(n)f FL(as)g(ar)-20 b(gument.)-913 7592 y FJ(CYCLOTOMIC)430 b FL(returns)f(the)h(c)-17 b(yclotomic)431 b(polynomial)g(of)e(order)h Fz(n)p FL(.)800 b(This)429 b(is)f(a)-913 8920 y(polynomial)279 b(ha)-22 b(ving)278 b(the)g Fz(n)p FL(-th)f(pritmiti)-28 b(v)-17 b(e)277 b(roots)g(of)g(the)g(unity)h(as)e(zeros.)-913 10248 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b(when)e Fz(n)308 b Fw(=)f(4)278 b FL(the)f(fourth)g(roots)g(of)g(the)g(unity)h (are:)343 b FI(f)p Fw(1)p Fz(;)184 b(i;)g FI(\000)p Fw(1)p Fz(;)g FI(\000)p Fz(i)p FI(g)p FL(;)-913 11577 y(among)279 b(them,)e(the)g(primiti)-28 b(v)-17 b(e)278 b(roots)f(are:)343 b FI(f)p Fz(i;)184 b FI(\000)p Fz(i)p FI(g)p FL(.)-913 12905 y(Therefore,)440 b(the)407 b(c)-17 b(yclotomic)409 b(polynomial)g(of)d(order)h Fw(4)g FL(is)f Fw(\()p Fz(X)429 b FI(\000)342 b Fz(i)p Fw(\))p Fz(:)p Fw(\()p Fz(X)431 b Fw(+)342 b Fz(i)p Fw(\))549 b(=)-913 14233 y Fz(X)91 13832 y Fv(2)833 14233 y Fw(+)246 b(1)p FL(.)-913 15562 y(Another)278 b(e)-17 b(xample,)279 b(typing:)11705 18048 y FA(CYCLOTOMIC)p Fw(\()p FA(20)p Fw(\))-913 20533 y FL(gi)-28 b(v)-17 b(es)278 b(:)10529 21937 y FA(X)11110 21480 y Fo(8)11823 21937 y FI(\000)246 b FA(X)13511 21480 y Fo(6)14225 21937 y Fw(+)f FA(X)15912 21480 y Fo(4)16626 21937 y FI(\000)h FA(X)18314 21480 y Fo(2)19027 21937 y Fw(+)g FA(1)-913 23762 y SDict begin H.S end -913 23762 a -913 23762 a SDict begin 12 H.A end -913 23762 a -913 23762 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.4) cvn H.B /DEST pdfmark end -913 23762 a 1483 x FE(5.8.4)1329 b(degr)-24 b(ee)331 b(of)h(a)h(polynomial)f(:)412 b Fq(DEGREE)-913 27369 y FJ(DEGREE)277 b FL(has)h(as)f(an)g(ar)-20 b(gument)279 b(a)e(polynomial)i(in)e(the)h(current)f(v)-28 b(ariable.)-913 28697 y FJ(DEGREE)277 b FL(returns)g(the)h(de)-17 b(gree)279 b(of)e(this)f(polynomial.)-885 30025 y(N)56 b FB(O)21 b(T)56 b(E)28 b FL(:)338 b(The)278 b(de)-17 b(gree)279 b(of)e(a)g(null)g(polynomial)i(is)d FJ(-1)p FL(.)-913 31354 y(The)i FJ(DEGREE)f FL(command)j(must)d(be)g(entered)i(using)e (the)h FJ(ALPHA)g FL(k)-11 b(e)-17 b(ys.)-913 32682 y(T)-89 b(yping:)10990 34086 y FA(DEGREE)p Fw(\()p FA(X)15487 33629 y Fo(2)16201 34086 y Fw(+)246 b FA(X)h Fw(+)f FA(1)p Fw(\))-913 36154 y FL(returns:)15331 37558 y FA(2)-913 39291 y SDict begin H.S end -913 39291 a -913 39291 a SDict begin 12 H.A end -913 39291 a -913 39291 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.5) cvn H.B /DEST pdfmark end -913 39291 a 1575 x FE(5.8.5)1329 b(Quotient)332 b(and)g(r)-24 b(emainder)330 b(:)412 b Fq(DIV2)-913 42990 y FL(The)278 b(ar)-20 b(guments)278 b(of)f FJ(DIV2)g FL(are)h(tw)-11 b(o)277 b(polynomials)h Fz(A)p Fw([)p Fz(X)87 b Fw(])278 b FL(and)g Fz(B)56 b Fw([)p Fz(X)87 b Fw(])p FL(.)-913 44318 y FJ(DIV2)278 b FL(returns)g(the)g(quotient)i FJ(AND)e FL(the)h(remainder)g(of)f(the)g(euclidean)j(di)-28 b(vision)279 b(be-)-913 45646 y(tween)f(the)f(tw)-11 b(o)277 b(polynomials)i Fz(A)p Fw([)p Fz(X)87 b Fw(])278 b FL(and)g Fz(B)56 b Fw([)p Fz(X)87 b Fw(])p FL(.)-913 46975 y(T)-89 b(yping:)10344 48379 y FA(DIV2)p Fw(\()p FA(X)13679 47922 y Fo(2)14393 48379 y Fw(+)246 b FA(2)h FI(\001)f FA(X)g Fw(+)g FA(1)p Fz(;)184 b FA(X)p Fw(\))-913 50447 y FL(gi)-28 b(v)-17 b(es:)12925 51851 y FA(X)247 b Fw(+)f FA(2)276 b(AND)i(1)-913 53919 y FL(The)256 b(step-by-step)f(mode)h(can)g(be)g(of)e(interest)h (here,)260 b(because)d(it)d(displays)h(the)g(inter)-22 b(-)-913 55248 y(mediate)278 b(steps)f(of)f(the)i(di)-28 b(vision)278 b(process.)p Black Black eop end end %%Page: 94 94 TeXDict begin HPSdict begin 94 93 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.94) cvn H.B /DEST pdfmark end -913 1926 a Black FL(94)661 b FD(CHAPTER)279 b(5.)620 b FJ(CAS)277 b FD(REFERENCE)j(FR)-44 b(OM)278 b(THE)f(EQ)-11 b(U)-44 b(A)-123 b(TION)278 b(EDIT)-20 b(OR)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.6) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(5.8.6)1329 b(B\351zout')-49 b(s)331 b(Identity)h(:)412 b Fq(EGCD)-913 6215 y FL(This)226 b(function)i(returns)e(B\351zout')-61 b(s)228 b(Identity)-72 b(,)237 b(the)227 b FJ(EGCD)g FL(\(Extended)i(Greatest)e(Com-)-913 7544 y(mon)274 b(Di)-28 b(visor\).)342 b(In)273 b(other)h(w)-11 b(ords,)274 b FJ(EGCD\(A\(X\),)666 b(B\(X\)\))274 b FL(returns)f FJ(U\(X\))665 b(AND)-913 8872 y(V\(X\))g(=)f(D\(X\))p FL(,)277 b(with)g FJ(D,)664 b(U,)h(V)277 b FL(such)h(that:)8350 11529 y FA(D)p Fw(\()p FA(X)p Fw(\))309 b(=)e FA(U)p Fw(\()p FA(X)p Fw(\))247 b FI(\001)g FA(A)p Fw(\()p FA(X)p Fw(\))g(+)f FA(V)p Fw(\()p FA(X)p Fw(\))h FI(\001)g FA(B)p Fw(\()p FA(X)p Fw(\))-913 13324 y FL(T)-89 b(yping:)9144 14652 y FA(EGCD)p Fw(\()p FA(X)12479 14195 y Fo(2)13193 14652 y Fw(+)246 b FA(2)g FI(\001)g FA(X)h Fw(+)f FA(1)p Fz(;)184 b FA(X)19268 14195 y Fo(2)19982 14652 y FI(\000)246 b FA(1)p Fw(\))-913 16448 y FL(gi)-28 b(v)-17 b(es:)10253 17776 y FA(1)277 b(AND)523 b FI(\000)246 b FA(1)585 b Fw(=)f FA(2)246 b FI(\001)h FA(X)f Fw(+)g FA(2)-913 19571 y FL(T)-89 b(yping:)9144 20900 y FA(EGCD)p Fw(\()p FA(X)12479 20443 y Fo(2)13193 20900 y Fw(+)246 b FA(2)g FI(\001)g FA(X)h Fw(+)f FA(1)p Fz(;)184 b FA(X)19268 20443 y Fo(3)19982 20900 y Fw(+)246 b FA(1)p Fw(\))-913 22695 y FL(gi)-28 b(v)-17 b(es:)9102 24023 y FI(\000)p Fw(\()p FA(X)246 b FI(\000)g FA(2)p Fw(\))278 b FA(AND)f(1)584 b Fw(=)g FA(3)247 b FI(\001)f FA(X)g Fw(+)g FA(3)-913 25860 y SDict begin H.S end -913 25860 a -913 25860 a SDict begin 12 H.A end -913 25860 a -913 25860 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.7) cvn H.B /DEST pdfmark end -913 25860 a 1273 x FE(5.8.7)1329 b(F)-33 b(actorize)331 b(a)i(polynomial)f (:)412 b Fq(FACTOR)-913 29208 y FJ(FACTOR)277 b FL(has)h(a)f (polynomial)i(as)e(an)g(ar)-20 b(gument.)-913 30537 y FJ(FACTOR)277 b FL(f)-11 b(actors)277 b(this)f(polynomial.)-913 31865 y(T)-89 b(yping:)11957 33193 y FA(FACTOR)p Fw(\()p FA(X)16454 32736 y Fo(2)17169 33193 y FI(\000)245 b FA(2)p Fw(\))-913 34989 y FL(gi)-28 b(v)-17 b(es:)10924 36317 y Fw(\()p FA(X)246 b Fw(+)13288 35366 y FI(p)p 14211 35366 582 45 v 14211 36317 a FA(2)p Fw(\))g FI(\001)h Fw(\()p FA(X)f FI(\000)18386 35366 y(p)p 19309 35366 V 19309 36317 a FA(2)p Fw(\))-913 38112 y FL(T)-89 b(yping:)10300 39441 y FA(FACTOR)p Fw(\()p FA(X)14797 38984 y Fo(2)15511 39441 y Fw(+)246 b FA(2)g FI(\001)h FA(X)f Fw(+)g FA(1)p Fw(\))-913 41236 y FL(gi)-28 b(v)-17 b(es:)13700 42564 y Fw(\()p FA(X)247 b Fw(+)f FA(1)p Fw(\))17076 42107 y Fo(2)-913 44360 y FL(T)-89 b(yping:)10066 45688 y FA(FACTOR)p Fw(\()p FA(X)14563 45231 y Fo(4)15278 45688 y FI(\000)246 b FA(2)g FI(\001)g FA(X)18346 45231 y Fo(2)19059 45688 y Fw(+)g FA(1)p Fw(\))-913 47483 y FL(gi)-28 b(v)-17 b(es:)11379 48812 y Fw(\()p FA(X)247 b FI(\000)f FA(1)p Fw(\))14755 48355 y Fo(2)15468 48812 y FI(\001)h Fw(\()p FA(X)f Fw(+)g FA(1)p Fw(\))19397 48355 y Fo(2)-913 50607 y FL(T)-89 b(yping:)10066 51935 y FA(FACTOR)p Fw(\()p FA(X)14563 51478 y Fo(3)15278 51935 y FI(\000)246 b FA(2)g FI(\001)g FA(X)18346 51478 y Fo(2)19059 51935 y Fw(+)g FA(1)p Fw(\))-913 53731 y FL(gi)-28 b(v)-17 b(es:)5091 55068 y Fw(\()p FA(X)246 b FI(\000)g FA(1)p Fw(\))h FI(\001)f Fw(\()p FA(2)h FI(\001)f FA(X)h FI(\000)e FA(1)i Fw(+)14946 54171 y FI(p)p 15868 54171 V 897 x FA(5)p Fw(\))g FI(\001)f Fw(\()p FA(2)h FI(\001)f FA(X)g FI(\000)g Fw(\()p FA(1)h Fw(+)23789 54171 y FI(p)p 24711 54171 V 897 x FA(5)q Fw(\)\))p 5091 55562 21063 45 v 15331 56576 a FA(4)p Black Black eop end end %%Page: 95 95 TeXDict begin HPSdict begin 95 94 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.95) cvn H.B /DEST pdfmark end -913 1926 a Black FD(5.8.)620 b(POL)-111 b(YNOMIALS)21865 b FL(95)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.8) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(5.8.8)1329 b(GCD)332 b(of)h(tw)-13 b(o)332 b(polynomials)g(:)412 b Fq(GCD)-913 6215 y FJ(GCD)277 b FL(returns)g(the)g(GCD)h(\(Greatest)e(Common)j(Di) -28 b(visor\))277 b(of)g(tw)-11 b(o)277 b(polynomials.)-913 7544 y(T)-89 b(yping:)9434 8872 y FA(GCD)p Fw(\()p FA(X)12188 8415 y Fo(2)12902 8872 y Fw(+)246 b FA(2)g FI(\001)h FA(X)f Fw(+)g FA(1)p Fz(;)184 b FA(X)18977 8415 y Fo(2)19691 8872 y FI(\000)246 b FA(1)p Fw(\))-913 10859 y FL(gi)-28 b(v)-17 b(es:)14364 12187 y FA(X)247 b Fw(+)f FA(1)-913 13897 y SDict begin H.S end -913 13897 a -913 13897 a SDict begin 12 H.A end -913 13897 a -913 13897 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.9) cvn H.B /DEST pdfmark end -913 13897 a 1457 x FE(5.8.9)1329 b(Hermite)332 b(polynomial)g(:)412 b Fq(HERMITE)-913 17430 y FJ(HERMITE)278 b FL(has)f(as)g(an)g(ar)-20 b(gument)279 b(a)e(whole)i(number)f Fz(n)p FL(.)-913 18758 y FJ(HERMITE)337 b FL(returns)g(the)g(Hermite)f(polynomial)j(of)e (de)-17 b(gree)338 b Fz(n)p FL(.)523 b(This)336 b(is)g(a)h(polyno-)-913 20087 y(mial)277 b(of)f(the)i(follo)-28 b(wing)278 b(type:)8492 23043 y Fz(H)9412 23209 y Ft(n)10015 23043 y Fw(\()p Fz(x)p Fw(\))308 b(=)g(\()p FI(\000)p Fw(1\))15259 22586 y Ft(n)16108 23043 y FI(\001)247 b Fz(e)17310 22289 y Fh(x)17748 22065 y Fm(2)p 17310 22412 871 38 v 17557 22853 a(2)18818 22294 y Fz(d)19394 21893 y Ft(n)p 18501 22789 1812 45 v 18501 23803 a Fz(dx)19710 23483 y Ft(n)20446 23043 y Fz(e)21094 22244 y Fg(\000)p Fh(x)22132 22020 y Fm(2)p 21094 22412 1470 38 v 21640 22853 a(2)-913 25322 y FL(That)277 b(means)h(that)g(:)-913 26651 y(for)e Fz(n)308 b FI(\025)f Fw(0)8038 27979 y Fz(H)9048 27188 y Fg(00)8958 28253 y Ft(n)9647 27979 y Fw(\()p Fz(x)p Fw(\))247 b FI(\000)f Fz(xH)14137 27188 y Fg(0)14047 28253 y Ft(n)14649 27979 y Fw(\()p Fz(x)p Fw(\))h(+)f Fz(nH)19080 28145 y Ft(n)19683 27979 y Fw(\()p Fz(x)p Fw(\))308 b(=)f(0)-913 29966 y FL(and)278 b(for)e Fz(n)308 b FI(\025)g Fw(1)6954 32392 y Fz(H)7874 32558 y Ft(n)p Fv(+1)9598 32392 y Fw(\()p Fz(x)p Fw(\))247 b FI(\000)e Fz(xH)13997 32558 y Ft(n)14600 32392 y Fw(\()p Fz(x)p Fw(\))h(+)g Fz(nH)19030 32558 y Ft(n)p Fs(\000)p Fv(1)20767 32392 y Fw(\()p Fz(x)p Fw(\))308 b(=)f(0)10969 34976 y Fz(H)11979 34185 y Fg(0)11889 35250 y Ft(n)12492 34976 y Fw(\()p Fz(x)p Fw(\))h(=)f Fz(nH)17045 35142 y Ft(n)p Fs(\000)p Fv(1)18782 34976 y Fw(\()p Fz(x)p Fw(\))-913 36963 y FL(T)-89 b(yping:)12867 38292 y FA(HERMITE)p Fw(\()p FA(6)p Fw(\))-913 40279 y FL(produces)278 b(the)g(result:)7625 42705 y FA(64)246 b FI(\001)h FA(X)10168 42248 y Fo(6)10881 42705 y FI(\000)f FA(480)g FI(\001)h FA(X)15112 42248 y Fo(4)15825 42705 y Fw(+)f FA(720)g FI(\001)h FA(X)20056 42248 y Fo(2)20769 42705 y FI(\000)f FA(120)-913 44414 y SDict begin H.S end -913 44414 a -913 44414 a SDict begin 12 H.A end -913 44414 a -913 44414 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.10) cvn H.B /DEST pdfmark end -913 44414 a 1458 x FE(5.8.10)1329 b(T)-122 b(o)332 b(e)-20 b(v)-13 b(aluate)332 b(a)g(polynomial)h(:)412 b Fq(HORNER)-913 47947 y FJ(HORNER)277 b FL(has)h(tw)-11 b(o)277 b(ar)-20 b(guments)278 b(:)343 b(a)277 b(polynomial)i FJ(P[X])f FL(and)g(a)f(number)i FJ(a)p FL(.)-913 49276 y FJ(HORNER\(P[X],a\))357 b FL(returns)e(the)h (polynomial)h FJ(Q[X])f FL(\(quotient)g(of)f FJ(P[X])h FL(di-)-913 50604 y(vided)278 b(by)g FJ(X-a)p FL(\))f FJ(AND)g FL(the)g(v)-28 b(alue)279 b FJ(P[a])p FL(.)-913 51932 y(T)-89 b(yping)279 b(:)11420 53261 y FA(HORNER)p Fw(\()p FA(X)15917 52804 y Fo(3)16632 53261 y Fw(+)246 b FA(1)p Fz(;)184 b FA(2)p Fw(\))-913 55248 y FL(gi)-28 b(v)-17 b(es)278 b(:)11035 56576 y FA(X)11616 56119 y Fo(2)12329 56576 y Fw(+)246 b FA(2)g FI(\001)g FA(X)g Fw(+)g FA(4)277 b(AND)g(9)p Black Black eop end end %%Page: 96 96 TeXDict begin HPSdict begin 96 95 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.96) cvn H.B /DEST pdfmark end -913 1926 a Black FL(96)661 b FD(CHAPTER)279 b(5.)620 b FJ(CAS)277 b FD(REFERENCE)j(FR)-44 b(OM)278 b(THE)f(EQ)-11 b(U)-44 b(A)-123 b(TION)278 b(EDIT)-20 b(OR)p Black -913 4140 a FL(T)-89 b(yping)279 b(:)6672 5865 y FA(HORNER)p Fw(\()p FA(X)11169 5408 y Fo(4)11883 5865 y Fw(+)246 b FA(2)g FI(\001)h FA(X)14952 5408 y Fo(3)15665 5865 y FI(\000)f FA(3)g FI(\001)g FA(X)18733 5408 y Fo(2)19447 5865 y Fw(+)g FA(X)g FI(\000)g FA(2)p Fz(;)184 b FA(1)p Fw(\))-913 8255 y FL(gi)-28 b(v)-17 b(es)278 b(:)10125 9980 y FA(X)10706 9523 y Fo(3)11419 9980 y Fw(+)246 b FA(3)g FI(\001)g FA(X)14487 9523 y Fo(2)15200 9980 y Fw(+)g FA(1)277 b(AND)523 b FI(\000)246 b FA(1)-913 12287 y SDict begin H.S end -913 12287 a -913 12287 a SDict begin 12 H.A end -913 12287 a -913 12287 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.11) cvn H.B /DEST pdfmark end -913 12287 a 1590 x FE(5.8.11)1329 b(LCM)332 b(of)h(tw)-13 b(o)332 b(polynomials)g(:)412 b Fq(LCM)-913 16204 y FJ(LCM)277 b FL(returns)g(the)g(LCM)h(\(Least)f (Common)h(Multiple\))f(of)g(tw)-11 b(o)277 b(polynomials.)-913 17532 y(T)-89 b(yping:)9434 19258 y FA(LCM)p Fw(\()p FA(X)12188 18801 y Fo(2)12902 19258 y Fw(+)246 b FA(2)g FI(\001)h FA(X)f Fw(+)g FA(1)p Fz(;)184 b FA(X)18977 18801 y Fo(2)19691 19258 y FI(\000)246 b FA(1)p Fw(\))-913 21647 y FL(gi)-28 b(v)-17 b(es:)9955 23373 y Fw(\()p FA(X)10966 22916 y Fo(2)11680 23373 y Fw(+)246 b FA(2)g FI(\001)g FA(X)h Fw(+)f FA(1)p Fw(\))g FI(\001)h Fw(\()p FA(X)f FI(\000)g FA(1)p Fw(\))-913 25864 y SDict begin H.S end -913 25864 a -913 25864 a SDict begin 12 H.A end -913 25864 a -913 25864 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.12) cvn H.B /DEST pdfmark end -913 25864 a 1405 x FE(5.8.12)1329 b(Legendr)-24 b(e)330 b(polynomial)j(:)412 b Fq(LEGENDRE)-913 29596 y FJ(LEGENDRE)278 b FL(has)f(as)g(an)h(ar)-20 b(gument)278 b(a)g(whole)g(number)g Fz(n)p FL(.)-913 30925 y FJ(LEGENDRE)273 b FL(returns)g(the)g(polynomial)i Fz(L)15713 31091 y Ft(n)16316 30925 y FL(,)e(a)f(non-null)i(solution)f (of)g(the)g(dif)-28 b(feren-)-913 32253 y(tial)276 b(equation:)5754 35307 y Fw(\()p Fz(x)6817 34850 y Fv(2)7560 35307 y FI(\000)246 b Fw(1\))h FI(\001)f Fz(y)11033 34850 y Fs(00)11844 35307 y FI(\000)g Fw(2)h FI(\001)f Fz(x)g FI(\001)g Fz(y)16319 34850 y Fs(0)16875 35307 y FI(\000)g Fz(n)p Fw(\()p Fz(n)i Fw(+)e(1\))h FI(\001)f Fz(y)347 b Fw(=)308 b(0)-913 37696 y FL(F)-17 b(or)277 b Fz(n)308 b FI(\025)g Fw(0)p FL(,)277 b(we)g(ha)-22 b(v)-17 b(e)279 b(the)e(Rodrigu\350s)i(F)-17 b(ormula:)8974 40937 y Fz(L)9727 41103 y Ft(n)10330 40937 y Fw(\()p Fz(x)p Fw(\))308 b(=)14220 40188 y(1)p 13433 40682 2128 45 v 13433 41696 a Fz(n)p Fw(!2)14957 41376 y Ft(n)16143 40188 y Fz(d)16719 39786 y Ft(n)p 15826 40682 1812 45 v 15826 41696 a Fz(dx)17035 41376 y Ft(n)17771 40937 y Fw(\()p Fz(x)18834 40480 y Fv(2)19576 40937 y FI(\000)246 b Fw(1\))21666 40480 y Ft(n)-913 43929 y FL(and)278 b(for)e Fz(n)308 b FI(\025)g Fw(1)p FL(,)277 b(we)g(ha)-22 b(v)-17 b(e:)4449 46629 y Fw(\()p Fz(n)247 b Fw(+)f(1\))p Fz(L)8633 46795 y Ft(n)p Fv(+1)10358 46629 y Fw(\()p Fz(x)p Fw(\))308 b(=)f(\(2)p Fz(n)248 b Fw(+)e(1\))p Fz(xL)18698 46795 y Ft(n)19301 46629 y Fw(\()p Fz(x)p Fw(\))h FI(\000)f Fz(nL)23565 46795 y Ft(n)p Fs(\000)p Fv(1)25301 46629 y Fw(\()p Fz(x)p Fw(\))-913 49329 y FL(T)-89 b(yping:)12576 51055 y FA(LEGENDRE)p Fw(\()p FA(4)p Fw(\))-913 53444 y FL(gi)-28 b(v)-17 b(es:)10969 55068 y FA(35)246 b FI(\001)g FA(X)13511 54666 y Fo(4)14225 55068 y FI(\000)f FA(30)i FI(\001)f FA(X)17874 54666 y Fo(2)18587 55068 y Fw(+)g FA(3)p 10969 55562 9307 45 v 15331 56576 a(8)p Black Black eop end end %%Page: 97 97 TeXDict begin HPSdict begin 97 96 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.97) cvn H.B /DEST pdfmark end -913 1926 a Black FD(5.8.)620 b(POL)-111 b(YNOMIALS)21865 b FL(97)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.13) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FE(5.8.13)1329 b(P)-13 b(artial)332 b(fraction)h(expansion)e(:)412 b Fq(PARTFRAC)-913 6232 y FJ(PARTFRAC)278 b FL(has)f(as)g(an)h(ar)-20 b(gument)278 b(a)g(rational)f(fraction.)-913 7560 y FJ(PARTFRAC)284 b FL(returns)f(the)h(partial)g(fraction)g(decomposition)i(of)d(this)g (rational)h(frac-)-913 8889 y(tion.)-913 10217 y(Example)279 b(:)-913 11545 y(T)-89 b(o)330 b(perform)g(a)f(partial)h(fraction)f (decomposition)j(of)e(a)f(rational)h(function,)344 b(for)329 b(e)-17 b(x-)-913 12874 y(ample:)11910 14152 y Fz(x)12543 13750 y Fv(5)13285 14152 y FI(\000)246 b Fw(2)h FI(\002)f Fz(x)16932 13750 y Fv(3)17674 14152 y Fw(+)g(1)p 7770 14646 15705 45 v 7770 15660 a Fz(x)8403 15341 y Fv(4)9145 15660 y FI(\000)g Fw(2)g FI(\002)g Fz(x)12791 15341 y Fv(3)13533 15660 y Fw(+)g(2)h FI(\002)f Fz(x)17180 15341 y Fv(2)17922 15660 y FI(\000)g Fw(2)h FI(\002)e Fz(x)h Fw(+)g(1)-913 17304 y FL(you)278 b(use)f(the)h FJ(PARTFRAC)g FL(command.)-913 18632 y(T)-89 b(ype:)5435 20634 y FA(PARTFRAC)p Fw(\()14442 19885 y FA(X)15023 19484 y Fo(5)15736 19885 y FI(\000)246 b FA(2)g FI(\003)g FA(X)19050 19484 y Fo(3)19764 19885 y Fw(+)f FA(1)p 10646 20380 14599 45 v 10646 21393 a(X)11227 21074 y Fo(4)11942 21393 y FI(\000)h FA(2)g FI(\003)g FA(X)15256 21074 y Fo(3)15969 21393 y Fw(+)g FA(2)g FI(\003)h FA(X)19284 21074 y Fo(2)19997 21393 y FI(\000)f FA(2)g FI(\003)h FA(X)f Fw(+)g FA(1)25379 20634 y Fw(\))-913 23037 y FL(In)276 b(real)i(and)g(direct)f(mode,)h (this)e(produces:)8616 25938 y FA(X)247 b Fw(+)f FA(2)g Fw(+)13541 25189 y FA(X)g FI(\000)g FA(3)p 12618 25684 4363 45 v 12618 26698 a(2)g FI(\001)g FA(X)14579 26378 y Fo(2)15292 26698 y Fw(+)g FA(2)17359 25938 y Fw(+)19826 25189 y FI(\000)p FA(1)p 18599 25684 3896 45 v 18599 26698 a(2)g FI(\001)h FA(X)f FI(\000)g FA(2)-913 28775 y FL(In)276 b(comple)-17 b(x)280 b(mode,)e(this)e(produces:)8164 31980 y FA(X)246 b Fw(+)g FA(2)g Fw(+)12327 30731 y Fo(1)p Fs(\000)p Fo(3)p Fs(\001)p Fo(i)p 12327 30912 2190 45 v 13216 31548 a(4)p 12165 31726 2516 45 v 12165 32740 a FA(X)g Fw(+)g FA(i)15059 31980 y Fw(+)17004 30731 y Fs(\000)p Fo(1)p 17004 30912 1104 45 v 17350 31548 a(2)p 16299 31726 2516 45 v 16299 32740 a FA(X)g FI(\000)g FA(1)19193 31980 y Fw(+)20601 30731 y Fo(1)p Fv(+)p Fo(3)p Fs(\001)p Fo(i)p 20601 30912 2178 45 v 21484 31548 a(4)p 20432 31726 2516 45 v 20432 32740 a FA(X)h FI(\000)e FA(i)-913 34490 y SDict begin H.S end -913 34490 a -913 34490 a SDict begin 12 H.A end -913 34490 a -913 34490 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.14) cvn H.B /DEST pdfmark end -913 34490 a 1034 x FE(5.8.14)1329 b(Whole-number)330 b(and)i(fractional)g(part)g(:)413 b Fq(PROPFRAC)-913 37617 y FJ(PROPFRAC)278 b FL(has)f(as)g(an)h(ar)-20 b(gument)278 b(a)g(rational)f(fraction.)-913 38945 y FJ(PROPFRAC)470 b FL(re)-28 b(writes)468 b(this)h(rational)g(fraction)h(so)f(as)f(to)h (bring)h(out)g(its)e(whole-)-913 40273 y(number)278 b(part.)-913 41743 y FJ(PROPFRAC\(A\(X\)/)666 b(B\(X\)\))248 b FL(writes)d(the)j (rational)f(fraction)24638 41206 y Ft(A)p Fv([)p Ft(X)60 b Fv(])p 24618 41488 1994 45 v 24618 42124 a Ft(B)38 b Fv([)p Ft(X)60 b Fv(])26991 41743 y FL(in)247 b(the)g(form:)12309 45132 y Fz(Q)p Fw([)p Fz(X)87 b Fw(])247 b(+)16312 44383 y Fz(R)9 b Fw([)p Fz(X)87 b Fw(])p 16289 44878 2515 45 v 16289 45892 a Fz(B)56 b Fw([)p Fz(X)87 b Fw(])-913 48234 y FL(where)278 b Fz(R)9 b Fw([)p Fz(X)87 b Fw(])307 b(=)h(0)p FL(,)277 b(or)f Fw(0)308 b FI(\024)g Fz(deg)40 b Fw(\()p Fz(R)9 b Fw([)p Fz(X)87 b Fw(]\))309 b Fz(<)e(deg)40 b Fw(\()p Fz(B)56 b Fw([)p Fz(X)87 b Fw(]\))p FL(.)-913 49562 y(T)-89 b(yping:)8268 51519 y FA(PROPFRAC)p Fw(\()13479 50770 y(\()p FA(5)248 b FI(\001)f FA(X)f Fw(+)g FA(3)p Fw(\))g FI(\001)h Fw(\()p FA(X)f FI(\000)g FA(1)p Fw(\))p 13479 51264 8933 45 v 16689 52278 a FA(X)g Fw(+)g FA(2)22546 51519 y Fw(\))-913 53922 y FL(gi)-28 b(v)-17 b(es:)11317 55725 y FA(5)246 b FI(\001)g FA(X)g FI(\000)g FA(12)h Fw(+)17956 54976 y FA(21)p 17279 55470 2516 45 v 17279 56484 a(X)g Fw(+)f FA(2)p Black Black eop end end %%Page: 98 98 TeXDict begin HPSdict begin 98 97 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a 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FL(in)g(order)h(of)e(its)g(po)-28 b(wers)278 b(of)f Fz(X)333 b FI(\000)246 b Fz(a)p FL(.)-913 8872 y(T)-89 b(yping:)10054 10200 y FA(PTAYL)p Fw(\()p FA(X)13970 9743 y Fo(2)14684 10200 y Fw(+)246 b FA(2)g FI(\001)h FA(X)f Fw(+)g FA(1)p Fz(;)184 b FA(2)p Fw(\))-913 12054 y FL(produces)278 b(the)g(polynomial)h(Q[X]:)12474 14257 y FA(X)13055 13800 y Fo(2)13768 14257 y Fw(+)246 b FA(6)g FI(\001)g FA(X)g Fw(+)g FA(9)-885 16461 y FL(N)56 b FB(O)21 b(T)56 b(E)328 b(T)56 b(H)g(A)-43 b(T)28 b FL(:)11894 17789 y FA(P)p Fw(\()p FA(X)p Fw(\))309 b(=)e FA(Q)p Fw(\()p FA(X)247 b FI(\000)f FA(2)p Fw(\))-913 19643 y SDict begin H.S end -913 19643 a -913 19643 a SDict begin 12 H.A end -913 19643 a -913 19643 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.16) cvn H.B /DEST pdfmark end -913 19643 a 1273 x FE(5.8.16)1329 b(Euclidean)331 b(quotient:)412 b Fq(QUOT)-913 22992 y FL(The)278 b(ar)-20 b(guments)278 b(of)f FJ(QUOT)g FL(are)h(tw)-11 b(o)277 b(polynomials)h Fz(A)p Fw([)p Fz(X)87 b Fw(])278 b FL(and)g Fz(B)56 b Fw([)p Fz(X)87 b Fw(])p FL(.)-913 24320 y FJ(QUOT)283 b FL(returns)f(the)h(quotient)h(of)f(the)g(tw)-11 b(o)282 b(polynomials)j Fz(A)p Fw([)p Fz(X)87 b Fw(])283 b FL(and)h Fz(B)56 b Fw([)p Fz(X)87 b Fw(])282 b FL(di)-28 b(vided)-913 25649 y(in)277 b(decreasing)i(order)e(by)h(e)-17 b(xponent.)-913 26977 y(T)-89 b(yping:)10344 28305 y FA(QUOT)p Fw(\()p FA(X)13679 27848 y Fo(2)14393 28305 y Fw(+)246 b FA(2)h FI(\001)f FA(X)g Fw(+)g FA(1)p Fz(;)184 b FA(X)p Fw(\))-913 30159 y FL(gi)-28 b(v)-17 b(es:)14364 31487 y FA(X)247 b Fw(+)f FA(2)-913 33157 y SDict begin H.S end -913 33157 a -913 33157 a SDict begin 12 H.A end -913 33157 a -913 33157 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.17) cvn H.B /DEST pdfmark end -913 33157 a 1457 x FE(5.8.17)1329 b(Euclidean)331 b(r)-24 b(emainder)330 b(:)412 b Fq(REMAINDER)-913 36690 y FL(The)278 b(ar)-20 b(guments)278 b(of)f FJ(REMAINDER)h FL(are)g(tw)-11 b(o)277 b(polynomials)h Fz(A)p Fw([)p Fz(X)87 b Fw(])278 b FL(and)g Fz(B)56 b Fw([)p Fz(X)87 b Fw(])p FL(.)-913 38018 y FJ(REMAINDER)261 b FL(returns)f(the)g (remainder)i(from)d(the)i(di)-28 b(vision)260 b(of)g(the)h(tw)-11 b(o)260 b(polynomi-)-913 39346 y(als)276 b Fz(A)p Fw([)p Fz(X)87 b Fw(])278 b FL(and)g Fz(B)56 b Fw([)p Fz(X)87 b Fw(])277 b FL(di)-28 b(vided)279 b(in)e(decreasing)i(order)e(by)h(e) -17 b(xponent.)-913 40675 y(T)-89 b(yping:)9348 42003 y FA(REMAINDER)p Fw(\()p FA(X)15588 41546 y Fo(3)16303 42003 y FI(\000)246 b FA(1)p Fz(;)184 b FA(X)19063 41546 y Fo(2)19777 42003 y FI(\000)246 b FA(1)p Fw(\))-913 43857 y FL(gi)-28 b(v)-17 b(es:)14364 45185 y FA(X)247 b FI(\000)f FA(1)-913 46854 y SDict begin H.S end -913 46854 a -913 46854 a SDict begin 12 H.A end -913 46854 a -913 46854 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.8.18) cvn H.B /DEST pdfmark end -913 46854 a 1458 x FE(5.8.18)1329 b(Number)330 b(of)j(zer)-24 b(os)331 b(in)h Fl([)p Fu(a;)221 b(b)p Fl([)333 b FE(:)412 b Fq(STURMAB)-913 50387 y FJ(STURMAB)377 b FL(has)f(three)h(ar)-20 b(guments:)543 b(a)376 b(polynomial)i Fz(P)154 b Fw(\()p Fz(X)87 b Fw(\))377 b FL(and)g(tw)-11 b(o)376 b(numbers)h Fz(a)-913 51716 y FL(and)278 b Fz(b)p FL(.)-913 53044 y FJ(STURMAB)g FL(returns)e(the)i(number)g(of)f(zeros)g (of)g Fz(P)430 b FL(in)277 b Fw([)p Fz(a;)184 b(b)p Fw([)p FL(.)-913 54373 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b(typing:)8809 56576 y FA(STURMAB)p Fw(\()p FA(X)13887 56119 y Fo(2)14601 56576 y FI(\001)247 b Fw(\()p FA(X)16166 56119 y Fo(3)16879 56576 y Fw(+)f FA(2)p Fw(\))p Fz(;)184 b FI(\000)p FA(2)p Fz(;)g FA(0)p Fw(\))p Black Black eop end end %%Page: 99 99 TeXDict begin HPSdict begin 99 98 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 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Fw(\()p Fz(x)p Fw(\))g FI(\000)g Fz(xT)15796 34161 y Fg(0)15642 35226 y Ft(n)16244 34952 y Fw(\()p Fz(x)p Fw(\))h(+)f Fz(n)19755 34495 y Fv(2)20252 34952 y Fz(T)20899 35118 y Ft(n)21501 34952 y Fw(\()p Fz(x)p Fw(\))309 b(=)e(0)-913 37415 y FL(F)-17 b(or)277 b Fz(n)308 b FI(\025)g Fw(1)p FL(,)277 b(we)g(ha)-22 b(v)-17 b(e:)8373 39878 y Fz(T)9020 40044 y Ft(n)p Fv(+1)10743 39878 y Fw(\()p Fz(x)p Fw(\))308 b(=)g(2)p Fz(xT)15546 40044 y Ft(n)16148 39878 y Fw(\()p Fz(x)p Fw(\))247 b FI(\000)f Fz(T)19642 40044 y Ft(n)p Fs(\000)p Fv(1)21377 39878 y Fw(\()p Fz(x)p Fw(\))-913 42342 y FL(If)271 b(n)i(<)f(0,)h FJ(TCHEBYCHEFF)i FL(returns)d(the)h (2nd-species)h(Tchebychef)-28 b(f)276 b(polynomial:)9433 45407 y Fz(T)10080 45573 y Ft(n)10683 45407 y Fw([)p Fz(x)p Fw(])308 b(=)13539 44658 y(sin\()p Fz(n)247 b FI(\001)g Fw(arccos\()p Fz(x)p Fw(\)\))p 13539 45152 8139 45 v 14271 46166 a(sin\(arccos)q(\()p Fz(x)p Fw(\)\))-913 48439 y FL(T)-89 b(yping:)11705 49809 y FA(TCHEBYCHEFF)p Fw(\()p FA(4)p 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H.V] /Dest (subsection.5.9.11) cvn H.B /DEST pdfmark end -913 43893 a 1309 x FE(5.9.11)1329 b(Ev)-13 b(aluate)332 b(a)g(primiti)-13 b(v)g(e)332 b(:)412 b Fq(PREVAL)-913 47278 y FJ(PREVAL)436 b FL(has)f(three)h(parameters:) 661 b(an)436 b(e)-17 b(xpression)437 b FJ(F\(VX\))f FL(dependent)j(on)d (the)-913 48606 y(v)-28 b(ariable)278 b(contained)i(in)d FJ(VX)p FL(,)g(and)h(tw)-11 b(o)277 b(e)-17 b(xpressions)278 b FJ(A)f FL(and)h FJ(B)p FL(.)-913 49934 y(F)-17 b(or)211 b(e)-17 b(xample)213 b(if)d FJ(VX)g FL(contains)i FJ(X)p FL(,)e(and)i(if)e FJ(F)g FL(is)g(a)h(function,)224 b FJ(PREVAL)666 b(\(F\(X\),A,B\))-913 51263 y FL(returns)276 b FJ(F\(B\)-F\(A\))p FL(.)-913 52591 y FJ(PREVAL)223 b FL(is)f(used)h(for)f(calculating)j(an)e(inte)-17 b(gral)224 b(de\002ned)h(from)e(a)g(primiti)-28 b(v)-17 b(e:)316 b(it)222 b(e)-28 b(v)g(al-)-913 53919 y(uates)277 b(this)f(primiti)-28 b(v)-17 b(e)278 b(between)h(the)f(tw)-11 b(o)277 b(limits)e(of)i(the)g (inte)-17 b(gral.)-913 55248 y(T)-89 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b(a)g(function)g Fz(f)396 b FL(of)277 b(the)g(current)h(v)-28 b(ariable)278 b FJ(VX)p FL(.)-913 46806 y FJ(SIGMAVX)288 b FL(returns)f(the)g(discrete)h(antideri)-28 b(v)g(ati)g(v)-17 b(e)290 b(of)d(the)g(input)h(function,)j(that)c(is)f(a)-913 48134 y(function)278 b Fz(G)f FL(that)g(satis\002es)f(the)i(relation:) 343 b Fz(G)p Fw(\()p Fz(x)247 b Fw(+)f(1\))h FI(\000)f Fz(G)p Fw(\()p Fz(x)p Fw(\))308 b(=)g Fz(f)119 b Fw(\()p Fz(x)p Fw(\))p FL(.)-913 49462 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b(type:)12633 50791 y FA(SIGMAVX)p Fw(\()p FA(X)17711 50334 y Fo(2)18180 50791 y Fw(\))-913 52587 y FL(Y)-122 b(ou)278 b(obtain:)11550 53588 y FA(2)246 b FI(\001)g FA(X)13511 53187 y Fo(3)14225 53588 y FI(\000)f FA(3)i FI(\001)f FA(X)17293 53187 y Fo(2)18006 53588 y Fw(+)g FA(X)p 11550 54083 8145 45 v 15331 55097 a(6)-913 56576 y FL(because)279 b Fw(2\()p Fz(X)334 b Fw(+)246 b(1\))7191 56174 y Fv(3)7934 56576 y FI(\000)g Fw(3\()p Fz(X)334 b Fw(+)246 b(1\))13365 56174 y Fv(2)14109 56576 y Fw(+)g Fz(X)333 b Fw(+)245 b(1)i FI(\000)f Fw(2)p Fz(X)21036 56174 y Fv(3)21779 56576 y Fw(+)g(3)p Fz(X)24443 56174 y Fv(2)25186 56576 y FI(\000)g Fz(X)394 b Fw(=)308 b(6)p Fz(X)30330 56174 y Fv(2)p Black Black eop end end %%Page: 112 112 TeXDict begin HPSdict begin 112 111 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.112) cvn H.B /DEST pdfmark end -913 1926 a Black FL(112)107 b FD(CHAPTER)280 b(5.)620 b FJ(CAS)277 b FD(REFERENCE)j(FR)-44 b(OM)278 b(THE)f(EQ)-11 b(U)-44 b(A)-123 b(TION)278 b(EDIT)-20 b(OR)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 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Fo(X)p Fv(+)p Fo(1)p 941 17920 1503 45 v 1487 18557 a(X)-913 19503 y FJ(Variation)665 b(table)p FL(:)7571 20958 y Fp(\024)8709 21843 y FI(\0001)1107 b Fw(?)1814 b(0)g(+)1107 b(+)p FI(1)g Fz(X)9432 23172 y Fw(?)1829 b(?)1107 b FI(\0001)1260 b(")h Fw(+)p FI(1)1176 b Fz(F)23089 20958 y Fp(\025)-913 25695 y SDict begin H.S end -913 25695 a -913 25695 a SDict begin 12 H.A end -913 25695 a -913 25695 a SDict begin [ /View [/XYZ H.V] /Dest (section.5.10) cvn H.B /DEST pdfmark end -913 25695 a 1293 x FF(5.10)1593 b(T)-147 b(aylor)399 b(and)g(asymptotic)f(Expansions)-913 29623 y FL(All)323 b(functions)h(in)f(this)g(section)h(can)h(be)f (found)h(in)f(the)g FJ(DIFF)g FL(menu)h(on)f(the)g(menu)-913 30952 y(bar)-61 b(.)-913 32280 y(It)329 b(is)g(customary)j(to)e(write)f (the)i(e)-17 b(xpansions)333 b(in)d(ascending)i(order)f(by)g(e)-17 b(xponent)333 b(of)-913 33608 y(the)277 b(v)-28 b(ariable;)278 b(you)h(set)d(this)g(up)i(by)g(choosing)h FA(1)246 b Fw(+)g FA(x)g Fw(+)g FA(x)21649 33207 y 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FL(\).)-913 25332 y SDict begin H.S end -913 25332 a -913 25332 a SDict begin 12 H.A end -913 25332 a -913 25332 a SDict begin [ /View [/XYZ H.V] /Dest (section.5.15) cvn H.B /DEST pdfmark end -913 25332 a 1716 x FF(5.15)1593 b(T)-118 b(rigonometric)398 b(Expr)-29 b(essions)-913 29522 y FL(All)347 b(the)i(functions)g(in)f (this)f(section)i(are)f(found)i(in)e(the)h FJ(TRIG)f FL(menu)i(on)e(the)h FJ(CAS)-913 30850 y FL(menu)278 b(bar)-61 b(.)-913 32433 y SDict begin H.S end -913 32433 a -913 32433 a SDict begin 12 H.A end -913 32433 a -913 32433 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.15.1) cvn H.B /DEST pdfmark end -913 32433 a 1534 x FE(5.15.1)1329 b(T)-98 b(ransf)-33 b(orm)330 b(the)i(ar)-24 b(ccos)332 b(into)g(ar)-24 b(csin)332 b(:)412 b Fq(ACOS2S)-913 36042 y FJ(ACOS2S)277 b FL(has)h(as)f(an)g(ar)-20 b(gument)279 b(a)e(trigonometric)h(e)-17 b(xpression.)-913 37371 y FJ(ACOS2S)277 b FL(transforms)g(the)g(e)-17 b(xpression)279 b(by)e(replacing)i(:)-913 38948 y Fw(arccos\()p Fz(x)p Fw(\))e FL(with)6196 38199 y Fz(\031)p 6196 38693 671 45 v 6255 39707 a Fw(2)7245 38948 y FI(\000)246 b Fw(arcsin\()p Fz(x)p Fw(\))p FL(.)-913 40568 y(T)-89 b(yping:)9005 41897 y FA(ACOS2S)p Fw(\()p FA(ACOS)p Fw(\()p FA(X)p Fw(\))249 b(+)d FA(ASIN)p Fw(\()p FA(X)p Fw(\)\))-913 43717 y FL(gi)-28 b(v)-17 b(es,)278 b(when)g(simpli\002ed)g(:)15287 44545 y Fz(\031)p 15287 45039 V 15332 46053 a FA(2)-913 47620 y SDict begin H.S end -913 47620 a -913 47620 a SDict begin 12 H.A end -913 47620 a -913 47620 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.15.2) cvn H.B /DEST pdfmark end -913 47620 a 1026 x FE(5.15.2)1329 b(T)-98 b(ransf)-33 b(orm)330 b(the)i(ar)-24 b(csin)332 b(into)g(ar)-24 b(ccos)332 b(:)412 b Fq(ASIN2C)-913 50722 y FJ(ASIN2C)277 b FL(has)h(as)f(an)g(ar) -20 b(gument)279 b(a)e(trigonometric)h(e)-17 b(xpression.)-913 52050 y FJ(ASIN2C)277 b FL(transforms)g(the)g(e)-17 b(xpression)279 b(by)e(replacing)i(:)-913 53627 y Fw(arcsin\()p Fz(x)p Fw(\))e 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Fp(\022)17246 24294 y FA(1)g FI(\000)g FA(SQ)p Fw(\()p FA(TAN)p Fw(\()23078 23858 y Fo(X)p 23080 24039 V 23080 24676 a(2)23624 24294 y Fw(\)\))p 17246 24853 7240 45 v 17246 25895 a FA(SQ)p Fw(\()p FA(TAN)p Fw(\()21144 25459 y Fo(X)p 21146 25640 412 45 v 21146 26277 a(2)21690 25895 y Fw(\)\))h(+)f FA(1)24618 23547 y Fp(\023)25433 23658 y Fo(2)-913 28166 y FL(or)-44 b(,)276 b(after)h (simpli\002cation:)15331 29495 y FA(1)-913 31099 y SDict begin H.S end -913 31099 a -913 31099 a SDict begin 12 H.A end -913 31099 a -913 31099 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.15.7) cvn H.B /DEST pdfmark end -913 31099 a 1550 x FE(5.15.7)1329 b(T)-98 b(ransf)-33 b(orm)465 b(the)g(complex)h(exponentials)f(into)i(sin)e(and)3736 34199 y(cos)332 b(:)412 b Fq(SINCOS)-913 36274 y FJ(SINCOS)263 b FL(tak)-11 b(es)263 b(as)f(an)h(ar)-20 b(gument)264 b(an)f(e)-17 b(xpression)264 b(containing)h(comple)-17 b(x)265 b(e)-17 b(xponen-)-913 37603 y(tials.)-913 38931 y FJ(SINCOS)277 b FL(then)h(re)-28 b(writes)277 b(this)f(e)-17 b(xpression)279 b(in)e(terms)f(of)h Fw(sin\()p Fz(x)p Fw(\))g FL(and)h Fw(cos)q(\()p Fz(x)p Fw(\))p FL(.)-913 40260 y(T)-89 b(yping:)11165 41588 y FA(SINCOS)p Fw(\()p FA(EXP)p Fw(\()p FA(i)248 b FI(\001)f FA(X)p Fw(\)\))-913 43533 y FL(gi)-28 b(v)-17 b(es)278 b(after)f(turning)g(on)h(comple)-17 b(x)279 b(mode,)g(if)d(necessary)i(:)11070 46190 y FA(COS)p Fw(\()p FA(X)p Fw(\))247 b(+)f FA(i)g FI(\001)h FA(SIN)p Fw(\()p FA(X)p Fw(\))-913 48071 y SDict begin H.S end -913 48071 a -913 48071 a SDict begin 12 H.A end -913 48071 a -913 48071 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.15.8) cvn H.B /DEST pdfmark end -913 48071 a 1273 x FE(5.15.8)1329 b(T)-98 b(ransf)-33 b(orm)251 b(tan\(x\))i(with)g(sin\(2x\))g(and)e (cos\(2x\))i(:)373 b Fq(TAN2CS2)-913 51420 y FJ(TAN2CS2)278 b FL(tak)-11 b(es)277 b(as)g(an)h(ar)-20 b(gument)278 b(a)g(trigonometric)g(e)-17 b(xpression.)-913 53350 y FJ(TAN2CS2)211 b FL(transforms)f(this)g(e)-17 b(xpression)212 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end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.164) cvn H.B /DEST pdfmark end -913 1926 a Black FL(164)16261 b FD(CHAPTER)279 b(7.)620 b(PR)-44 b(OGRAMMING)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (section.7.10) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FF(7.10)1593 b(WHILE)399 b(statement)-913 5397 y SDict begin H.S end -913 5397 a -913 5397 a SDict begin 12 H.A end -913 5397 a -913 5397 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.7.10.1) cvn H.B /DEST pdfmark end -913 5397 a 1633 x FE(7.10.1)1329 b(Algorithmic)332 b(syntax)-913 9299 y FJ(While)665 b(condition)g(do)g (action)g(endwhile)-913 11402 y SDict begin H.S end -913 11402 a -913 11402 a SDict begin 12 H.A end -913 11402 a -913 11402 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.7.10.2) cvn H.B /DEST pdfmark end -913 11402 a 1629 x FE(7.10.2)1329 b Fq(HP40GS)332 b FE(syntax)-913 15301 y FJ(WHILE)665 b Fc(condition)g FJ(REPEAT)h Fc(action)f FJ(:)1328 b(END:)-913 17528 y SDict begin H.S end -913 17528 a -913 17528 a SDict begin 12 H.A end -913 17528 a -913 17528 a SDict begin [ /View [/XYZ H.V] /Dest (section.7.11) cvn H.B /DEST pdfmark end -913 17528 a 2071 x FF(7.11)1593 b(Boolean)399 b(Expr)-29 b(essions)-913 22268 y FL(A)326 b(condition)i(is)e(a)g(function)i(that) f(has)f(a)h(Boolean)i(as)d(a)g(v)-28 b(alue,)341 b(that)326 b(is,)338 b(it)326 b(is)f(either)-913 23596 y FJ(TRUE)277 b FL(or)g FJ(FALSE)p FL(.)-913 25699 y SDict begin H.S end -913 25699 a -913 25699 a SDict begin 12 H.A end -913 25699 a -913 25699 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.7.11.1) cvn H.B /DEST pdfmark end -913 25699 a 1628 x FE(7.11.1)1329 b(Algorithmic)332 b(syntax)-913 29597 y FL(T)-89 b(o)278 b(e)-17 b(xpress)278 b(a)f(condition,)h(simply)f (use)h(the)f(operators:)747 31028 y FJ(=)664 b(>)h(>)f FI(\024)g(\025)g(6)p Fw(=)-913 33322 y SDict begin H.S end -913 33322 a -913 33322 a SDict begin 12 H.A end -913 33322 a -913 33322 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.7.11.2) cvn H.B /DEST pdfmark end -913 33322 a 1437 x FE(7.11.2)1329 b Fq(HP40GS)332 b FE(syntax)-885 37029 y FL(N)56 b FB(O)21 b(T)56 b(E)28 b FL(:)338 b(for)276 b(the)i FJ(HP40GS)f FL(calculator)-44 b(,)278 b(equality)h(is)d (denoted)j(by:)-913 38358 y FJ(==)-913 39686 y FL(Otherwise,)e(the)g (operators)h(are)f(the)h(same.)-913 42130 y SDict begin H.S end -913 42130 a -913 42130 a SDict begin 12 H.A end -913 42130 a -913 42130 a SDict begin [ /View [/XYZ H.V] /Dest (section.7.12) cvn H.B /DEST pdfmark end -913 42130 a 1855 x FF(7.12)1593 b(Logical)399 b(Operators)-913 45555 y SDict begin H.S end -913 45555 a -913 45555 a SDict begin 12 H.A end -913 45555 a -913 45555 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.7.12.1) cvn H.B /DEST pdfmark end -913 45555 a 1319 x FE(7.12.1)1329 b(Algorithmic)332 b(syntax)-913 49144 y FL(T)-89 b(o)278 b(e)-17 b(xpress)278 b(comple)-17 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(page.165) cvn H.B /DEST pdfmark end -913 1926 a Black FD(7.13.)620 b(LISTS)25626 b FL(165)p Black -913 3033 a SDict begin H.S end -913 3033 a -913 3033 a SDict begin 12 H.A end -913 3033 a -913 3033 a SDict begin [ /View [/XYZ H.V] /Dest (section.7.13) cvn H.B /DEST pdfmark end -913 3033 a 1107 x FF(7.13)1593 b(Lists)-913 5404 y SDict begin H.S end -913 5404 a -913 5404 a SDict begin 12 H.A end -913 5404 a -913 5404 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.7.13.1) cvn H.B /DEST pdfmark end -913 5404 a 1642 x FE(7.13.1)1329 b(Algorithmic)332 b(syntax)-913 9332 y FL(In)276 b(algorithms,)h(you)i(use)e FJ({)664 b(})277 b FL(to)g(de\002ne)i(a)e(list.)-913 10661 y(F)-17 b(or)386 b(e)-17 b(xample,)415 b FJ({})386 b FL(designates)h(a)f(v)-22 b(oid)387 b(list,)411 b(and)387 b({1,)413 b(2,)g(3})387 b(is)e(a)h(list)e(with)i(three)-913 11989 y(elements.)-913 13317 y(The)216 b FJ(+)e FL(is)g(used)i(to)e(concatenate)219 b(tw)-11 b(o)215 b(lists,)225 b(or)215 b(a)g(list)e(and)j(an)f (element,)229 b(or)214 b(an)i(element)-913 14646 y(and)278 b(a)f(list:)-913 15974 y FJ({1,)664 b(2,)h(3})f(->)h(TAB)-913 17302 y(TAB)f(+)h(4)f(->)h(TAB)277 b FL(\()p FJ(TAB)g FL(no)-28 b(w)278 b(designates)g FJ({1,)665 b(2,)f(3,)h(4})-913 18631 y(TAB[2])277 b FL(indicates)h(the)g(second)h(element)f(of)f FJ(TAB)p FL(,)g(here)g FJ(2)p FL(.)-913 20913 y SDict begin H.S end -913 20913 a -913 20913 a SDict begin 12 H.A end -913 20913 a -913 20913 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.7.13.2) cvn H.B /DEST pdfmark end -913 20913 a 1496 x FE(7.13.2)1329 b Fq(HP40GS)332 b FE(syntax)-913 24696 y FL(The)278 b(list)d(v)-28 b(ariables)278 b(ha)-22 b(v)-17 b(e)279 b(the)f(follo)-28 b(wing)278 b(names:)344 b FJ(L0,)665 b(L1,)f(L2,...)1330 b(L9)p FL(.)-913 26024 y(Y)-122 b(ou)278 b(use)f FJ({)664 b(})277 b FL(to)g(delimit)g(a)g (list.)-913 27352 y(F)-17 b(or)277 b(e)-17 b(xample,)280 b FJ({1,)664 b(2,)h(3})277 b FL(is)f(a)h(list)f(with)h(three)g (elements.)-913 28681 y(But)290 b FJ({})g 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FL(designates)i(the)e(Ith)g (element)h(of)f(the)g(list.)-913 41964 y FJ(CONCAT)665 b(\(L1,)g({5}\))332 b FL(designates)h(a)f(list)f(ha)-22 b(ving)334 b(the)e(element)h FJ(5)f FL(in)g(addition)-913 43293 y(to)277 b(the)g(elements)h(of)f(the)g(list)f FJ(L1)p FL(.)-913 44621 y(Y)-122 b(ou)278 b(can)g(also)f(use:)-913 45949 y FJ(AUGMENT\(L1,5\))p FL(,)349 b(which)335 b(designates)g(a)f (list)f(ha)-22 b(ving)335 b(the)f(element)h FJ(5)f FL(in)g(addi-)-913 47278 y(tion)277 b(to)g(the)g(elements)h(of)f(the)g(list)f FJ(L1)p FL(.)-913 48606 y FJ(SUB)664 b(L2;)h(L1;)g(2;)f(4)310 b FL(is)g(a)g(command)j(that)d(copies)h(into)f FJ(L2)h FL(the)f(elements)h(of)-913 49934 y FJ(L1)277 b FL(ha)-22 b(ving)278 b(indices)g(from)f(2)g(to)g(4.)-885 51263 y(N)56 b FB(O)21 b(T)56 b(E)300 b FL(the)277 b(dif)-28 b(ference)279 b(between)g(functions)e(and)i(commands:)-913 52591 y(Functions)363 b(return)e(a)h(v)-28 b(alue,)383 b(and)363 b(the)-17 b(y)363 b(ha)-22 b(v)-17 b(e)363 b(parentheses)g(enclosing)g(their)e(ar)-20 b(gu-)-913 53919 y(ments)277 b(\(which)h(are)f(separated)h(by)g(commas\),)g (whereas)-913 55248 y(Commands)369 b(do)g(not)f(return)g(v)-28 b(alues,)391 b(and)369 b(their)e(ar)-20 b(guments)369 b(are)f(written)g(directly)-913 56576 y(after)276 b(the)i(name)g(of)f (the)g(command)j(\(and)e(are)f(separated)i(by)e(semi-colons\).)p Black Black eop end end %%Page: 166 166 TeXDict begin HPSdict begin 166 165 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray -913 1926 a SDict begin H.S end -913 1926 a 0 TeXcolorgray 0 TeXcolorgray -913 1926 a SDict begin H.R end -913 1926 a -913 1926 a SDict begin [ /View [/XYZ H.V] /Dest (page.166) cvn H.B /DEST pdfmark end -913 1926 a Black FL(166)16261 b FD(CHAPTER)279 b(7.)620 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