GAUSS
to perform reduction of a quadratic form q.
There are two ways to use GAUSS
:
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'
Example:
The matrix of q defined by q(x,y)=4x2+2xy-3y2 is:
(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:
This means that:
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.