Quadratic forms.

The main program is GAUSS (located in the other directory) to perform reduction of a quadratic form q. There are two ways to use GAUSS:

• symbolic input:
  Input: a quadratic form q (symbolic) at level 1 or the quadratic form q at level 2 and the list of variables at level 1.
  Output:
  
  • stk5: D the list of diagonal coefficients (only the number of positive and negative coefficients is characteristic of q )
  • stk4: P (the columns vectors of P^-1 form a q-orthogonal basis of A at level 3)
  • stk3: A (A is the matrix of q in the dual base of the coordinates-forms at level 2, we have A=P^t D P where P^t denotes the transposed of P)
  • stk2: list of variables
  • stk1: symbolic as a sum of independent squares

  Examples:
  

  Example 1:
  'X^2+4*X*Y-2*X*Z+4*Y^2+6*Y*Z+7*Z^2' GAUSS
  5: { 1 '-25/6' '1/6' }
  4: { { 1 2 -1 } { 0 1 0 } { 0 5 6 } }
  3: { { 1 2 -1 } { 2 4 3 } { -1 3 7 } }
  2: { X Y Z }
  1: '1/6*(6*Z+5*Y)^2+ -25/6*Y^2+(-Z+2*Y+X)^2'
  
  Example 2: same example but with variable in the reverse order
  'X^2+4*X*Y-2*X*Z+4*Y^2+6*Y*Z+7*Z^2' { Z Y X } GAUSS
  5: { '1/7' '7/19' '-25/19' }
  4: { { 7 3 -1 } { 0 '19/7' '17/7' } { 0 0 1 } }
  3: { { 7 3 -1 } { 3 4 2 } { -1 2 1 } }
  2: { Z Y X }
  1: '-25/19*X^2+7/19*(17/7*X+19/7*Y)^2+1/7*(-X+3*Y+7*Z)^2
       
  Example 3: if you want to orthogonalize with parameter, you need to
  enter the list of variables of the quadratic form
  'X^2+2*A*X*Y' { Y X } GAUSS
  5: { '-A^2' 1 }
  4: { { 1 0 } { A 1 } }
  3: { { 0 A } { A 1 } }
  2: { Y X }
  1: '(X+A*Y)^2-A^2*Y^2'
  

• matrix input:
  Input (stack level 1): the formal matrix A of the quadratic form q
  Output: at stack level 2 D the diagonal coefficients list and at stack level 1 the transition matrix P. We have A=P^t D P where P^t denotes the transposed of P. Note that to obtain a q-orthogonal basis, one can take the columns of the inverse P^-1 of P).

  Example:
  The matrix of q defined by q(x,y)=4x²+2xy-3y² is:
  

  $$A=\left(\begin{array}{lcr} 4 & 1\\ 1 & -3 \end{array} \right) ,$$
  
  
  (to get the matrix of q, enter '4*X^2+2*X*Y-3*Y^2', then the list of variables { X Y} and hit QXA). Call GAUSS which returns:

  2: { '1/4' '-13/4' }
  1: { { 4 1 } { 0 1 } }
  

  this means that:
  
  $$A= \left(\begin{array}{lcr} 4 & 0\\ 1 & 1 \end{array} \ri... ...left(\begin{array}{lcr} 4 & 1\\ 0 & 1 \end{array} \right).$$
  
  
  This means that:
  
  $$q(x,y)=4x^{2}+2xy-3y^{2} = \frac{1}{4} (4x+y)^{2}-\frac{13}{4} y^{2}.$$
  
  

The other utilities are QXA and AXQ to switch from algebraic to matricial representation of a quadratic form (quadratic as symbolic to array). QXA accepts an optional list of variables at level 1.