ppequ#

Functions

void sppequ(
    const char*          uplo,
    const INT            n,
    const f32*  restrict AP,
          f32*  restrict S,
          f32*           scond,
          f32*           amax,
          INT*           info
);
void sppequ(const char *uplo, const INT n, const f32 *restrict AP, f32 *restrict S, f32 *scond, f32 *amax, INT *info)#

SPPEQU computes row and column scalings intended to equilibrate a symmetric positive definite matrix A in packed storage 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.

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
AP

Array of dimension n*(n+1)/2. The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1.

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 dppequ(
    const char*          uplo,
    const INT            n,
    const f64*  restrict AP,
          f64*  restrict S,
          f64*           scond,
          f64*           amax,
          INT*           info
);
void dppequ(const char *uplo, const INT n, const f64 *restrict AP, f64 *restrict S, f64 *scond, f64 *amax, INT *info)#

DPPEQU computes row and column scalings intended to equilibrate a symmetric positive definite matrix A in packed storage 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.

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
AP

Array of dimension n*(n+1)/2. The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1.

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 cppequ(
    const char*          uplo,
    const INT            n,
    const c64*  restrict AP,
          f32*  restrict S,
          f32*           scond,
          f32*           amax,
          INT*           info
);
void cppequ(const char *uplo, const INT n, const c64 *restrict AP, f32 *restrict S, f32 *scond, f32 *amax, INT *info)#

CPPEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A in packed storage 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.

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
AP

Array of dimension n*(n+1)/2. The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1.

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 zppequ(
    const char*          uplo,
    const INT            n,
    const c128* restrict AP,
          f64*  restrict S,
          f64*           scond,
          f64*           amax,
          INT*           info
);
void zppequ(const char *uplo, const INT n, const c128 *restrict AP, f64 *restrict S, f64 *scond, f64 *amax, INT *info)#

ZPPEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A in packed storage 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.

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
AP

Array of dimension n*(n+1)/2. The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1.

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.