Computes row and column scaling factors restricted to a power of radix to equilibrate a banded matrix and reduce its condition number.

## Syntax

lapack_int LAPACKE_sgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const float* ab, lapack_int ldab, float* r, float* c, float* rowcnd, float* colcnd, float* amax );

lapack_int LAPACKE_dgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const double* ab, lapack_int ldab, double* r, double* c, double* rowcnd, double* colcnd, double* amax );

lapack_int LAPACKE_cgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const lapack_complex_float* ab, lapack_int ldab, float* r, float* c, float* rowcnd, float* colcnd, float* amax );

lapack_int LAPACKE_zgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int kl, lapack_int ku, const lapack_complex_double* ab, lapack_int ldab, double* r, double* c, double* rowcnd, double* colcnd, double* amax );

• mkl.h

## Description

The routine computes row and column scalings intended to equilibrate an m-by-n banded matrix A and reduce its condition number. The output array r returns the row scale factors and the array c - the column scale factors. These factors are chosen to try to make the largest element in each row and column of the matrix B with elements bi, j=r[i-1]*ai, j*c[j-1] have an absolute value of at most the radix.

r[i] and c[j] are restricted to be a power of the radix between SMLNUM = smallest safe number and BIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of a but works well in practice.

SMLNUM and BIGNUM are parameters representing machine precision. You can use the ?lamch routines to compute them. For example, compute single precision values of SMLNUM and BIGNUM as follows:

```SMLNUM = slamch ('s')
BIGNUM = 1 / SMLNUM```

This routine differs from ?gbequ by restricting the scaling factors to a power of the radix. Except for over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries' magnitudes are no longer equal to approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

## Input Parameters

 matrix_layout Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR). m The number of rows of the matrix A; m≥ 0. n The number of columns of the matrix A; n≥ 0. kl The number of subdiagonals within the band of A; kl≥ 0. ku The number of superdiagonals within the band of A; ku≥ 0. ab Array: size max(1, ldab*n) for column major layout and max(1, ldab*m) for row major layout ldab The leading dimension of a; ldab≥ max(1, m).

## Output Parameters

 r, c Arrays: r (size m), c (size n). If info = 0, or info>m, the array r contains the row scale factors for the matrix A. If info = 0, the array c contains the column scale factors for the matrix A. rowcnd If info = 0 or info>m, rowcnd contains the ratio of the smallest r(i) to the largest r(i). If rowcnd≥ 0.1, and amax is neither too large nor too small, it is not worth scaling by r. colcnd If info = 0, colcnd contains the ratio of the smallest c[i] to the largest c[i]. If colcnd≥ 0.1, it is not worth scaling by c. amax Absolute value of the largest element of the matrix A. If amax is very close to SMLNUM or BIGNUM, the matrix should be scaled.

## Return Values

This function returns a value info.

If info = 0, the execution is successful.

If info = -i, parameter i had an illegal value.

If info = i, the i-th diagonal element of A is nonpositive.

im, the i-th row of A is exactly zero;

i>m, the (i-m)-th column of A is exactly zero.