?tpcon

Estimates the reciprocal of the condition number of a packed triangular matrix.

Syntax

FORTRAN 77:

call stpcon( norm, uplo, diag, n, ap, rcond, work, iwork, info )

call dtpcon( norm, uplo, diag, n, ap, rcond, work, iwork, info )

call ctpcon( norm, uplo, diag, n, ap, rcond, work, rwork, info )

call ztpcon( norm, uplo, diag, n, ap, rcond, work, rwork, info )

FORTRAN 95:

call tpcon( ap, rcond [,uplo] [,diag] [,norm] [,info] )

lapack_int LAPACKE_stpcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, const float* ap, float* rcond );

lapack_int LAPACKE_dtpcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, const double* ap, double* rcond );

lapack_int LAPACKE_ctpcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, const lapack_complex_float* ap, float* rcond );

lapack_int LAPACKE_ztpcon( int matrix_order, char norm, char uplo, char diag, lapack_int n, const lapack_complex_double* ap, 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 triangular matrix A in either the 1-norm or infinity-norm:

κ₁(A) =||A||₁ ||A^-1||₁ = κ_∞(A^T) = κ_∞(A^H)

κ_∞(A) =||A||_∞ ||A^-1||_∞ =κ₁ (A^T) = κ₁(A^H) .

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.

norm	CHARACTER*1. Must be '1' or 'O' or 'I'. If `norm` = '1' or 'O', then the routine estimates the condition number of matrix `A` in 1-norm. If `norm = 'I'`, then the routine estimates the condition number of matrix `A` in infinity-norm.
uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates whether `A` is upper or lower triangular: If `uplo = 'U'`, the array `ap` stores the upper triangle of `A` in packed form. If `uplo = 'L'`, the array `ap` stores the lower triangle of `A` in packed form.
diag	CHARACTER*1. Must be 'N' or 'U'. If `diag = 'N'`, then `A` is not a unit triangular matrix. If `diag = 'U'`, then `A` is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array `ap`.
n	INTEGER. The order of the matrix `A`; `n` ≥ 0.
ap, work	REAL for stpcon DOUBLE PRECISION for dtpcon COMPLEX for ctpcon DOUBLE COMPLEX for ztpcon. Arrays: `ap()`, `work()`. The array `ap` contains the packed matrix `A`. 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.
iwork	INTEGER. Workspace array, DIMENSION at least `max(1, n)`.
rwork	REAL for ctpcon DOUBLE PRECISION for ztpcon. 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 tpcon interface are as follows:

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

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 n² floating-point operations for real flavors and 4n² operations for complex flavors.