(S|D|C|Z)GEBAL¶

Single, double, single complex, and double complex GEBAL.

Description¶

Balances a general matrix using permutations and scaling to isolate eigenvalues and make rows and columns more similar in norm.

The permutation stage puts the matrix A into the form:

$P \mult A \mult \herm{P} = \begin{bmatrix} T_1 & X & Y \\ 0 & B & Z \\ 0 & 0 & T_2 \end{bmatrix}$

where $T_1$ and $T_2$ are upper triangular. The indices ILO and IHI define the boundaries of B. Permutation details are recorded in SCALE in components 1 through ILO-1, and IHI+1 through N.

The balance stage consists of diagonal similarity transforms:

$\inv{D} \mult B \mult D$

The scale factors for D are chosen to make the 1-norm of each row of B and its corresding column as equal as possible. The scale factors for D are recorded in SCALE in components ILO through IHI.

The final balanced matrix is:

$\inv{D} \mult P \mult A \mult \herm{P} \mult D = \begin{bmatrix} T_1 & X \mult D & Y \\ 0 & \inv{D} \mult B \mult D & \inv{D} \mult Z \\ 0 & 0 & T_2 \end{bmatrix}$

LAPACK Interface¶

void sgebal(const char *JOB, const qml_long *N, float *A, const qml_long *LDA,
    qml_long *ILO, qml_long *IHI, float *SCALE, qml_long *INFO);

void dgebal(const char *JOB, const qml_long *N, double *A, const qml_long *LDA,
    qml_long *ILO, qml_long *IHI, double *SCALE, qml_long *INFO);

void cgebal(const char *JOB, const qml_long *N, qml_single_complex *A,
    const qml_long *LDA, qml_long *ILO, qml_long *IHI, float *SCALE,
    qml_long *INFO);

void zgebal(const char *JOB, const qml_long *N, qml_double_complex *A,
    const qml_long *LDA, qml_long *ILO, qml_long *IHI, double *SCALE,
    qml_long *INFO);

Arguments¶

JOB	Operations to perform, ‘N’ none, ‘P’ permute only, ‘S’ scale only, ‘B’ both
N	Number of rows and columns of A
A	Matrix of size N x N, overwritten with balanced matrix
LDA	Leading dimension of A
ILO	Lower index
IHI	Upper index
SCALE	Array of size N, scale factors of columns
INFO	0 on success