?ppcon

Estimates the reciprocal of the condition number of a packed symmetric (Hermitian) positive-definite matrix.

Syntax

FORTRAN 77:

call sppcon( uplo, n, ap, anorm, rcond, work, iwork, info )

call dppcon( uplo, n, ap, anorm, rcond, work, iwork, info )

call cppcon( uplo, n, ap, anorm, rcond, work, rwork, info )

call zppcon( uplo, n, ap, anorm, rcond, work, rwork, info )

FORTRAN 95:

call ppcon( ap, anorm, rcond [,uplo] [,info] )

lapack_int LAPACKE_sppcon( int matrix_order, char uplo, lapack_int n, const float* ap, float anorm, float* rcond );

lapack_int LAPACKE_dppcon( int matrix_order, char uplo, lapack_int n, const double* ap, double anorm, double* rcond );

lapack_int LAPACKE_cppcon( int matrix_order, char uplo, lapack_int n, const lapack_complex_float* ap, float anorm, float* rcond );

lapack_int LAPACKE_zppcon( int matrix_order, char uplo, lapack_int n, const lapack_complex_double* ap, double anorm, double* rcond );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine estimates the reciprocal of the condition number of a packed 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 = max_j Σ_i |a_ij| or ||A||_∞ = max_i Σ_j |a_ij|)
call ?pptrf to compute the Cholesky factorization of A.

Input Parameters

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.
ap, work	REAL for sppcon DOUBLE PRECISION for dppcon COMPLEX for cppcon DOUBLE COMPLEX for zppcon. Arrays: `ap()`, `work()`. The array `ap` contains the packed factored matrix `A`, as returned by ?pptrf. The dimension of `ap` must be at least max(1,`n`(`n`+1)/2). The array `work` is a workspace for the routine. The dimension of `work` must be at least `max(1, 3n)` for real flavors and `max(1, 2n)` for complex flavors.
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 cppcon DOUBLE PRECISION for zppcon. Workspace array, DIMENSION at least `max(1, n)`.

Output Parameters

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.

Fortran 95 Interface Notes

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 ppcon interface are as follows:

ap	Holds the array `A` of size `(n*(n+1)/2)`.
uplo	Must be 'U' or 'L'. The default value is 'U'.

Application Notes

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.