heequb#

Functions

void cheequb(
    const char*          uplo,
    const INT            n,
    const c64*  restrict A,
    const INT            lda,
          f32*  restrict S,
          f32*           scond,
          f32*           amax,
          c64*  restrict work,
          INT*           info
);
void cheequb(const char *uplo, const INT n, const c64 *restrict A, const INT lda, f32 *restrict S, f32 *scond, f32 *amax, c64 *restrict work, INT *info)#

CHEEQUB computes row and column scalings intended to equilibrate a Hermitian matrix A (with respect to the Euclidean norm) and reduce its condition number.

The scale factors S are computed by the BIN algorithm (see references) so that the scaled matrix B with elements B(i,j)=S(i)*A(i,j)*S(j) has a condition number within a factor N of the smallest possible condition number over all possible diagonal scalings.

References:

Livne, O.E. and Golub, G.H., “Scaling by Binormalization”, Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. https://doi.org/10.1023/B:NUMA.0000016606.32820.69

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The order of the matrix A. n>=0.

in
A

Array of dimension (lda,n). The n-by-n Hermitian matrix whose scaling factors are to be computed.

in
lda

The leading dimension of the array A. lda>=max(1,n).

out
S

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

out
scond

If info=0, S contains the ratio of the smallest S(i) to the largest S(i). If scond>=0.1 and amax is neither too large nor too small, it is not worth scaling by S.

out
amax

Largest absolute value of any matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out
work

Array of dimension 2*n.

out
info

  • info=0: successful exit

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

  • info>0: if info=i, the i-th diagonal element is nonpositive.

Functions

void zheequb(
    const char*          uplo,
    const INT            n,
    const c128* restrict A,
    const INT            lda,
          f64*  restrict S,
          f64*           scond,
          f64*           amax,
          c128* restrict work,
          INT*           info
);
void zheequb(const char *uplo, const INT n, const c128 *restrict A, const INT lda, f64 *restrict S, f64 *scond, f64 *amax, c128 *restrict work, INT *info)#

ZHEEQUB computes row and column scalings intended to equilibrate a Hermitian matrix A (with respect to the Euclidean norm) and reduce its condition number.

The scale factors S are computed by the BIN algorithm (see references) so that the scaled matrix B with elements B(i,j)=S(i)*A(i,j)*S(j) has a condition number within a factor N of the smallest possible condition number over all possible diagonal scalings.

References:

Livne, O.E. and Golub, G.H., “Scaling by Binormalization”, Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. https://doi.org/10.1023/B:NUMA.0000016606.32820.69

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The order of the matrix A. n>=0.

in
A

Array of dimension (lda,n). The n-by-n Hermitian matrix whose scaling factors are to be computed.

in
lda

The leading dimension of the array A. lda>=max(1,n).

out
S

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

out
scond

If info=0, S contains the ratio of the smallest S(i) to the largest S(i). If scond>=0.1 and amax is neither too large nor too small, it is not worth scaling by S.

out
amax

Largest absolute value of any matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out
work

Array of dimension 2*n.

out
info

  • info=0: successful exit

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

  • info>0: if info=i, the i-th diagonal element is nonpositive.