Computes row and column scaling factors intended to equilibrate a symmetric (Hermitian) positive definite matrix in packed storage and reduce its condition number.

Syntax

lapack_int LAPACKE_sppequ( int matrix_layout, char uplo, lapack_int n, const float* ap, float* s, float* scond, float* amax );

lapack_int LAPACKE_dppequ( int matrix_layout, char uplo, lapack_int n, const double* ap, double* s, double* scond, double* amax );

lapack_int LAPACKE_cppequ( int matrix_layout, char uplo, lapack_int n, const lapack_complex_float* ap, float* s, float* scond, float* amax );

lapack_int LAPACKE_zppequ( int matrix_layout, char uplo, lapack_int n, const lapack_complex_double* ap, double* s, double* scond, double* amax );

Include Files

  • mkl.h

Description

The routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). The output array s returns scale factors such that contains


Equation

These factors are chosen so that the scaled matrix B with elements bij=s[i-1]*aij*s[j-1] has diagonal elements equal to 1.

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.

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

uplo

Must be 'U' or 'L'.

Indicates whether the upper or lower triangular part of A is packed in the array ap:

If uplo = 'U', the array ap stores the upper triangular part of the matrix A.

If uplo = 'L', the array ap stores the lower triangular part of the matrix A.

n

The order of matrix A; n 0.

ap

Array, size at least max(1,n(n+1)/2). The array ap contains the upper or the lower triangular part of the matrix A (as specified by uplo) in packed storage (see Matrix Storage Schemes).

Output Parameters

s

Array, size (n).

If info = 0, the array s contains the scale factors for A.

scond

If info = 0, scond contains the ratio of the smallest s[i] to the largest s[i].

amax

Absolute value of the largest element of the matrix A.

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.

Application Notes

If scond 0.1 and amax is neither too large nor too small, it is not worth scaling by s.

If amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.

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