?gtcon

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?gtcon

Estimates the reciprocal of the condition number of a tridiagonal matrix.

Syntax

call sgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info )

call dgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info )

call cgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, info )

call zgtcon( norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, info )

call gtcon( dl, d, du, du2, ipiv, anorm, rcond [,norm] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine estimates the reciprocal of the condition number of a real or complex tridiagonal matrix A in the 1-norm or infinity-norm:

κ₁(A) = ||A||₁||A^-1||₁

κ_∞(A) = ||A||_∞||A^-1||_∞

An estimate is obtained for ||A^-1||, and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A^-1||).

Before calling this routine:

compute anorm (either ||A||₁ = max_jΣ_i |a_ij| or ||A||_∞ = max_iΣ_j |a_ij|)
call ?gttrf to compute the LU factorization of A.

Input Parameters

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.
n	INTEGER. The order of the matrix `A`; `n`≥ 0.
dl,d,du,du2	REAL for sgtcon DOUBLE PRECISION for dgtcon COMPLEX for cgtcon DOUBLE COMPLEX for zgtcon. Arrays: `dl(n -1)`, `d(n)`, `du(n -1)`, `du2(n -2)`. The array `dl` contains the `(n - 1)` multipliers that define the matrix `L` from the `LU` factorization of `A` as computed by ?gttrf. The array `d` contains the `n` diagonal elements of the upper triangular matrix `U` from the `LU` factorization of `A`. The array `du` contains the (`n` - 1) elements of the first superdiagonal of `U`. The array `du2` contains the (`n` - 2) elements of the second superdiagonal of `U`.
ipiv	INTEGER. Array, size (`n`). The array of pivot indices, as returned by ?gttrf.
anorm	REAL for single precision flavors. DOUBLE PRECISION for double precision flavors. The norm of the original matrix `A`(see Description).
work	REAL for sgtcon DOUBLE PRECISION for dgtcon COMPLEX for cgtcon DOUBLE COMPLEX for zgtcon. Workspace array, size (2*`n`).
iwork	INTEGER. Workspace array, size (`n`). Used for real flavors only.

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.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine gtcon interface are as follows:

dl	Holds the vector of length (`n`-1).
d	Holds the vector of length `n`.
du	Holds the vector of length (`n`-1).
du2	Holds the vector of length (`n`-2).
ipiv	Holds the vector of length `n`.
norm	Must be '1', 'O', or 'I'. The default value is '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 2n² floating-point operations for real flavors and 8n² for complex flavors.

Parent topic: Estimating the Condition Number: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?gtcon