poequb#

Functions

void spoequb(
    const INT           n,
    const f32* restrict A,
    const INT           lda,
          f32* restrict S,
          f32*          scond,
          f32*          amax,
          INT*          info
);
void spoequb(const INT n, const f32 *restrict A, const INT lda, f32 *restrict S, f32 *scond, f32 *amax, INT *info)#

SPOEQUB computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

This routine differs from SPOEQU 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 diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

Parameters

in
n

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

in
A

Array of dimension (lda, n). The n-by-n symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.

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

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, the i-th diagonal element is nonpositive.

Functions

void dpoequb(
    const INT           n,
    const f64* restrict A,
    const INT           lda,
          f64* restrict S,
          f64*          scond,
          f64*          amax,
          INT*          info
);
void dpoequb(const INT n, const f64 *restrict A, const INT lda, f64 *restrict S, f64 *scond, f64 *amax, INT *info)#

DPOEQUB computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

This routine differs from DPOEQU 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 diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

Parameters

in
n

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

in
A

Array of dimension (lda, n). The n-by-n symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.

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

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, the i-th diagonal element is nonpositive.

Functions

void cpoequb(
    const INT           n,
    const c64* restrict A,
    const INT           lda,
          f32* restrict S,
          f32*          scond,
          f32*          amax,
          INT*          info
);
void cpoequb(const INT n, const c64 *restrict A, const INT lda, f32 *restrict S, f32 *scond, f32 *amax, INT *info)#

CPOEQUB computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

This routine differs from CPOEQU 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 diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

Parameters

in
n

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

in
A

Array of dimension (lda, n). The n-by-n Hermitian positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.

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

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, the i-th diagonal element is nonpositive.

Functions

void zpoequb(
    const INT            n,
    const c128* restrict A,
    const INT            lda,
          f64*  restrict S,
          f64*           scond,
          f64*           amax,
          INT*           info
);
void zpoequb(const INT n, const c128 *restrict A, const INT lda, f64 *restrict S, f64 *scond, f64 *amax, INT *info)#

ZPOEQUB computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

This routine differs from ZPOEQU 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 diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).

Parameters

in
n

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

in
A

Array of dimension (lda, n). The n-by-n Hermitian positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.

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

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, the i-th diagonal element is nonpositive.