gbequb#

Functions

void sgbequb(
    const INT           m,
    const INT           n,
    const INT           kl,
    const INT           ku,
    const f32* restrict AB,
    const INT           ldab,
          f32* restrict R,
          f32* restrict C,
          f32*          rowcnd,
          f32*          colcnd,
          f32*          amax,
          INT*          info
);
void sgbequb(const INT m, const INT n, const INT kl, const INT ku, const f32 *restrict AB, const INT ldab, f32 *restrict R, f32 *restrict C, f32 *rowcnd, f32 *colcnd, f32 *amax, INT *info)#

SGBEQUB computes row and column scalings intended to equilibrate an m-by-n band matrix A and reduce its condition number.

R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) 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.

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

Parameters

in
m

The number of rows of the matrix A. m>=0.

in
n

The number of columns of the matrix A. n>=0.

in
kl

The number of subdiagonals within the band of A. kl>=0.

in
ku

The number of superdiagonals within the band of A. ku>=0.

in
AB

Array of dimension (ldab, n). The band matrix A, stored in band format. The matrix A is stored in rows 0 to kl+ku, so that AB[ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(m-1,j+kl).

in
ldab

The leading dimension of the array AB. ldab>=kl+ku+1.

out
R

Array of dimension m. If info=0 or info>m, R contains the row scale factors for A.

out
C

Array of dimension n. If info=0, C contains the column scale factors for A.

out
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.

out
colcnd

If info=0, colcnd contains the ratio of the smallest C(j) to the largest C(j). If colcnd>=0.1, it is not worth scaling by C.

out
amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, and i is

    • i<=m: the i-th row of A is exactly zero (1-based)

    • i>m: the (i-m)-th column of A is exactly zero (1-based)

Functions

void dgbequb(
    const INT           m,
    const INT           n,
    const INT           kl,
    const INT           ku,
    const f64* restrict AB,
    const INT           ldab,
          f64* restrict R,
          f64* restrict C,
          f64*          rowcnd,
          f64*          colcnd,
          f64*          amax,
          INT*          info
);
void dgbequb(const INT m, const INT n, const INT kl, const INT ku, const f64 *restrict AB, const INT ldab, f64 *restrict R, f64 *restrict C, f64 *rowcnd, f64 *colcnd, f64 *amax, INT *info)#

DGBEQUB computes row and column scalings intended to equilibrate an m-by-n band matrix A and reduce its condition number.

R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) 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.

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

Parameters

in
m

The number of rows of the matrix A. m>=0.

in
n

The number of columns of the matrix A. n>=0.

in
kl

The number of subdiagonals within the band of A. kl>=0.

in
ku

The number of superdiagonals within the band of A. ku>=0.

in
AB

Array of dimension (ldab, n). The band matrix A, stored in band format. The matrix A is stored in rows 0 to kl+ku, so that AB[ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(m-1,j+kl).

in
ldab

The leading dimension of the array AB. ldab>=kl+ku+1.

out
R

Array of dimension m. If info=0 or info>m, R contains the row scale factors for A.

out
C

Array of dimension n. If info=0, C contains the column scale factors for A.

out
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.

out
colcnd

If info=0, colcnd contains the ratio of the smallest C(j) to the largest C(j). If colcnd>=0.1, it is not worth scaling by C.

out
amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, and i is

    • i<=m: the i-th row of A is exactly zero (1-based)

    • i>m: the (i-m)-th column of A is exactly zero (1-based)

Functions

void cgbequb(
    const INT           m,
    const INT           n,
    const INT           kl,
    const INT           ku,
    const c64* restrict AB,
    const INT           ldab,
          f32* restrict R,
          f32* restrict C,
          f32*          rowcnd,
          f32*          colcnd,
          f32*          amax,
          INT*          info
);
void cgbequb(const INT m, const INT n, const INT kl, const INT ku, const c64 *restrict AB, const INT ldab, f32 *restrict R, f32 *restrict C, f32 *rowcnd, f32 *colcnd, f32 *amax, INT *info)#

CGBEQUB computes row and column scalings intended to equilibrate an m-by-n band matrix A and reduce its condition number.

R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) 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.

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

Parameters

in
m

The number of rows of the matrix A. m>=0.

in
n

The number of columns of the matrix A. n>=0.

in
kl

The number of subdiagonals within the band of A. kl>=0.

in
ku

The number of superdiagonals within the band of A. ku>=0.

in
AB

Array of dimension (ldab, n). The band matrix A, stored in band format. The matrix A is stored in rows 0 to kl+ku, so that AB[ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(m-1,j+kl).

in
ldab

The leading dimension of the array AB. ldab>=kl+ku+1.

out
R

Array of dimension m. If info=0 or info>m, R contains the row scale factors for A.

out
C

Array of dimension n. If info=0, C contains the column scale factors for A.

out
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.

out
colcnd

If info=0, colcnd contains the ratio of the smallest C(j) to the largest C(j). If colcnd>=0.1, it is not worth scaling by C.

out
amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, and i is

    • i<=m: the i-th row of A is exactly zero (1-based)

    • i>m: the (i-m)-th column of A is exactly zero (1-based)

Functions

void zgbequb(
    const INT            m,
    const INT            n,
    const INT            kl,
    const INT            ku,
    const c128* restrict AB,
    const INT            ldab,
          f64*  restrict R,
          f64*  restrict C,
          f64*           rowcnd,
          f64*           colcnd,
          f64*           amax,
          INT*           info
);
void zgbequb(const INT m, const INT n, const INT kl, const INT ku, const c128 *restrict AB, const INT ldab, f64 *restrict R, f64 *restrict C, f64 *rowcnd, f64 *colcnd, f64 *amax, INT *info)#

ZGBEQUB computes row and column scalings intended to equilibrate an m-by-n band matrix A and reduce its condition number.

R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) 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.

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

Parameters

in
m

The number of rows of the matrix A. m>=0.

in
n

The number of columns of the matrix A. n>=0.

in
kl

The number of subdiagonals within the band of A. kl>=0.

in
ku

The number of superdiagonals within the band of A. ku>=0.

in
AB

Array of dimension (ldab, n). The band matrix A, stored in band format. The matrix A is stored in rows 0 to kl+ku, so that AB[ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(m-1,j+kl).

in
ldab

The leading dimension of the array AB. ldab>=kl+ku+1.

out
R

Array of dimension m. If info=0 or info>m, R contains the row scale factors for A.

out
C

Array of dimension n. If info=0, C contains the column scale factors for A.

out
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.

out
colcnd

If info=0, colcnd contains the ratio of the smallest C(j) to the largest C(j). If colcnd>=0.1, it is not worth scaling by C.

out
amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, and i is

    • i<=m: the i-th row of A is exactly zero (1-based)

    • i>m: the (i-m)-th column of A is exactly zero (1-based)