Computes row and column scaling factors intended to equilibrate a symmetric (Hermitian) positive definite matrix and reduce its condition number.
FORTRAN 77:
call spoequb( n, a, lda, s, scond, amax, info )
call dpoequb( n, a, lda, s, scond, amax, info )
call cpoequb( n, a, lda, s, scond, amax, info )
call zpoequb( n, a, lda, s, scond, amax, info )
C:
lapack_int LAPACKE_spoequb( int matrix_order, lapack_int n, const float* a, lapack_int lda, float* s, float* scond, float* amax );
lapack_int LAPACKE_dpoequb( int matrix_order, lapack_int n, const double* a, lapack_int lda, double* s, double* scond, double* amax );
lapack_int LAPACKE_cpoequb( int matrix_order, lapack_int n, const lapack_complex_float* a, lapack_int lda, float* s, float* scond, float* amax );
lapack_int LAPACKE_zpoequb( int matrix_order, lapack_int n, const lapack_complex_double* a, lapack_int lda, double* s, double* scond, double* amax );
The routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive-definite matrix A and reduce its condition number (with respect to the two-norm).
These factors are chosen so that the scaled matrix B with elements b(i,j)=s(i)*a(i,j)*s(j) has diagonal elements equal to 1. s(i) is a power of two nearest to, but not exceeding 1/sqrt(A(i,i)).
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.
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
n |
INTEGER. The order of the matrix A; n ≥ 0. |
a |
REAL for spoequb DOUBLE PRECISION for dpoequb COMPLEX for cpoequb DOUBLE COMPLEX for zpoequb. Array: DIMENSION (lda,*). Contains the n-by-n symmetric or Hermitian positive definite matrix A whose scaling factors are to be computed. Only the diagonal elements of A are referenced. The second dimension of a must be at least max(1,n). |
lda |
INTEGER. The leading dimension of a; lda ≥ max(1, m). |
s |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, DIMENSION (n). If info = 0, the array s contains the scale factors for A. |
scond |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. If info = 0, scond 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. |
amax |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Absolute value of the largest element of the matrix A. If amax is very close to overflow or underflow, the matrix should be scaled. |
info |
INTEGER. If info = 0, the execution is successful. If info = -i, the i-th parameter had an illegal value. If info = i, the i-th diagonal element of A is nonpositive. |