echelon SciMax Toolbox eigenvalues

SciMax Toolbox >> eigens_by_jacobi

eigens_by_jacobi

Maxima Function

Calling Sequence

eigens_by_jacobi (A)
eigens_by_jacobi(A,field_type)

Description

Computes the eigenvalues and eigenvectors of A by the method of Jacobi rotations. A must be a symmetric matrix (but it need not be positive definite nor positive semidefinite). field_type indicates the computational field, either floatfield or bigfloatfield. If field_type is not specified, it defaults to floatfield.

The elements of A must be numbers or expressions which evaluate to numbers via float or bfloat (depending on field_type).

Examples:

(%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]);
                     [     1         1    ]
                     [  -------   ------- ]
                     [  sqrt(2)   sqrt(2) ]
(%o1)                [                    ]
                     [      1        1    ]
                     [ - -------  ------- ]
                     [   sqrt(2)  sqrt(2) ]
(%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]);
                      [ sqrt(3)     0    ]
(%o2)                 [                  ]
                      [    0     sqrt(5) ]
(%i3) M : S . L . transpose (S);
            [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
            [ ------- + -------  ------- - ------- ]
            [    2         2        2         2    ]
(%o3)       [                                      ]
            [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
            [ ------- - -------  ------- + ------- ]
            [    2         2        2         2    ]
(%i4) eigens_by_jacobi (M);
The largest percent change was 0.1454972243679
The largest percent change was 0.0
number of sweeps: 2
number of rotations: 1
(%o4) [[1.732050807568877, 2.23606797749979],
                        [  0.70710678118655   0.70710678118655 ]
                        [                                      ]]
                        [ - 0.70710678118655  0.70710678118655 ]
(%i5) float ([[sqrt(3), sqrt(5)], S]);
(%o5) [[1.732050807568877, 2.23606797749979],
                        [  0.70710678118655   0.70710678118655 ]
                        [                                      ]]
                        [ - 0.70710678118655  0.70710678118655 ]
(%i6) eigens_by_jacobi (M, bigfloatfield);
The largest percent change was 1.454972243679028b-1
The largest percent change was 0.0b0
number of sweeps: 2
number of rotations: 1
(%o6) [[1.732050807568877b0, 2.23606797749979b0],
                [  7.071067811865475b-1   7.071067811865475b-1 ]
                [                                              ]]
                [ - 7.071067811865475b-1  7.071067811865475b-1 ]
echelon SciMax Toolbox eigenvalues