Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite matrix.
FORTRAN 77:
call spocon( uplo, n, a, lda, anorm, rcond, work, iwork, info )
call dpocon( uplo, n, a, lda, anorm, rcond, work, iwork, info )
call cpocon( uplo, n, a, lda, anorm, rcond, work, rwork, info )
call zpocon( uplo, n, a, lda, anorm, rcond, work, rwork, info )
FORTRAN 95:
call pocon( a, anorm, rcond [,uplo] [,info] )
C:
lapack_int LAPACKE_spocon( int matrix_order, char uplo, lapack_int n, const float* a, lapack_int lda, float anorm, float* rcond );
lapack_int LAPACKE_dpocon( int matrix_order, char uplo, lapack_int n, const double* a, lapack_int lda, double anorm, double* rcond );
lapack_int LAPACKE_cpocon( int matrix_order, char uplo, lapack_int n, const lapack_complex_float* a, lapack_int lda, float anorm, float* rcond );
lapack_int LAPACKE_zpocon( int matrix_order, char uplo, lapack_int n, const lapack_complex_double* a, lapack_int lda, double anorm, double* rcond );
The routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite matrix A:
κ1(A) = ||A||1 ||A-1||1 (since A is symmetric or Hermitian, κ∞(A) = κ1(A)).
Before calling this routine:
compute anorm (either ||A||1 = maxj Σi |aij| or ||A||∞ = maxi Σj |aij|)
call ?potrf to compute the Cholesky factorization of A.
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.
uplo |
CHARACTER*1. Must be 'U' or 'L'. Indicates how the input matrix A has been factored: If uplo = 'U', the upper triangle of A is stored. If uplo = 'L', the lower triangle of A is stored. |
n |
INTEGER. The order of the matrix A; n ≥ 0. |
a, work |
REAL for spocon DOUBLE PRECISION for dpocon COMPLEX for cpocon DOUBLE COMPLEX for zpocon. Arrays: a(lda,*), work(*). The array a contains the factored matrix A, as returned by ?potrf. The second dimension of a must be at least max(1,n). The array work is a workspace for the routine. The dimension of work must be at least max(1, 3*n) for real flavors and max(1, 2*n) for complex flavors. |
lda |
INTEGER. The leading dimension of a; lda ≥ max(1, n). |
anorm |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. The norm of the original matrix A (see Description). |
iwork |
INTEGER. Workspace array, DIMENSION at least max(1, n). |
rwork |
REAL for cpocon DOUBLE PRECISION for zpocon. Workspace array, DIMENSION at least max(1, n). |
rcond |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. An estimate of the reciprocal of the condition number. The routine sets rcond =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular. |
info |
INTEGER. If info = 0, the execution is successful. If info = -i, the i-th parameter had an illegal value. |
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.
Specific details for the routine pocon interface are as follows:
a |
Holds the matrix A of size (n, n). |
uplo |
Must be 'U' or 'L'. The default value is 'U'. |
The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10r. A call to this routine involves solving a number of systems of linear equations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2 floating-point operations for real flavors and 8n2 for complex flavors.