?ptcon

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

Syntax

FORTRAN 77:

call sptcon( n, d, e, anorm, rcond, work, info )

call dptcon( n, d, e, anorm, rcond, work, info )

call cptcon( n, d, e, anorm, rcond, work, info )

call zptcon( n, d, e, anorm, rcond, work, info )

FORTRAN 95:

call ptcon( d, e, anorm, rcond [,info] )

lapack_int LAPACKE_sptcon( lapack_int n, const float* d, const float* e, float anorm, float* rcond );

lapack_int LAPACKE_dptcon( lapack_int n, const double* d, const double* e, double anorm, double* rcond );

lapack_int LAPACKE_cptcon( lapack_int n, const float* d, const lapack_complex_float* e, float anorm, float* rcond );

lapack_int LAPACKE_zptcon( lapack_int n, const double* d, const lapack_complex_double* e, 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 computes the reciprocal of the condition number (in the 1-norm) of a real symmetric or complex Hermitian positive-definite tridiagonal matrix using the factorization A = L*D*L^T for real flavors and A = L*D*L^H for complex flavors or A = U^T*D*U for real flavors and A = U^H*D*U for complex flavors computed by ?pttrf :

κ₁(A) = ||A||₁ ||A^-1||₁ (since A is symmetric or Hermitian, κ_∞(A) = κ₁(A)).

The norm ||A^-1|| is computed by a direct method, and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A^-1||).

Before calling this routine:

compute anorm as ||A||₁ = max_j Σ_i |a_ij|
call ?pttrf to compute the 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.

n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
d, work	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Arrays, dimension (`n`). The array `d` contains the `n` diagonal elements of the diagonal matrix `D` from the factorization of `A`, as computed by ?pttrf ; `work` is a workspace array.
e	REAL for sptcon DOUBLE PRECISION for dptcon COMPLEX for cptcon DOUBLE COMPLEX for zptcon. Array, DIMENSION `(n -1)`. Contains off-diagonal elements of the unit bidiagonal factor `U` or `L` from the factorization computed by ?pttrf .
anorm	REAL for single precision flavors. DOUBLE PRECISION for double precision flavors. The 1- norm of the original matrix `A` (see Description).

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 gtcon interface are as follows:

d	Holds the vector of length `n`.
e	Holds the vector of length (`n`-1).

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.