?pbcon

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

Syntax

FORTRAN 77:

call spbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info )

call dpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info )

call cpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info )

call zpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info )

FORTRAN 95:

call pbcon( ab, anorm, rcond [,uplo] [,info] )

lapack_int LAPACKE_spbcon( int matrix_order, char uplo, lapack_int n, lapack_int kd, const float* ab, lapack_int ldab, float anorm, float* rcond );

lapack_int LAPACKE_dpbcon( int matrix_order, char uplo, lapack_int n, lapack_int kd, const double* ab, lapack_int ldab, double anorm, double* rcond );

lapack_int LAPACKE_cpbcon( int matrix_order, char uplo, lapack_int n, lapack_int kd, const lapack_complex_float* ab, lapack_int ldab, float anorm, float* rcond );

lapack_int LAPACKE_zpbcon( int matrix_order, char uplo, lapack_int n, lapack_int kd, const lapack_complex_double* ab, lapack_int ldab, 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 symmetric (Hermitian) positive-definite band 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 ?pbtrf 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 triangular factor is stored in `ab`. If `uplo = 'L'`, the lower triangular factor is stored in `ab`.
n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
kd	INTEGER. The number of superdiagonals or subdiagonals in the matrix `A`; `kd` ≥ 0.
ldab	INTEGER. The leading dimension of the array `ab`. (`ldab` ≥ `kd` +1).
ab, work	REAL for spbcon DOUBLE PRECISION for dpbcon COMPLEX for cpbcon DOUBLE COMPLEX for zpbcon. Arrays: `ab(ldab,)`, `work()`. The array `ab` contains the factored matrix `A` in band form, as returned by ?pbtrf. The second dimension of `ab` 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, 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 cpbcon DOUBLE PRECISION for zpbcon. 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 pbcon interface are as follows:

ab	Holds the array `A` of size `(kd+1,n)`.
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 4*n(kd + 1) floating-point operations for real flavors and 16*n(kd + 1) for complex flavors.