f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_dgtcon (f07cgc)

1  Purpose

nag_dgtcon (f07cgc) estimates the reciprocal condition number of a real $n$ by $n$ tridiagonal matrix $A$, using the $LU$ factorization returned by nag_dgttrf (f07cdc).

2  Specification

 #include #include
 void nag_dgtcon (Nag_NormType norm, Integer n, const double dl[], const double d[], const double du[], const double du2[], const Integer ipiv[], double anorm, double *rcond, NagError *fail)

3  Description

nag_dgtcon (f07cgc) should be preceded by a call to nag_dgttrf (f07cdc), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals. nag_dgtcon (f07cgc) then utilizes the factorization to estimate either ${‖{A}^{-1}‖}_{1}$ or ${‖{A}^{-1}‖}_{\infty }$, from which the estimate of the reciprocal of the condition number of $A$, $1/\kappa \left(A\right)$ is computed as either
 $1 / κ1 A = 1 / A1 A-11$
or
 $1 / κ∞ A = 1 / A∞ A-1∞ .$
$1/\kappa \left(A\right)$ is returned, rather than $\kappa \left(A\right)$, since when $A$ is singular $\kappa \left(A\right)$ is infinite.
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.

4  References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5  Arguments

1:     normNag_NormTypeInput
On entry: specifies the norm to be used to estimate $\kappa \left(A\right)$.
${\mathbf{norm}}=\mathrm{Nag_OneNorm}$
Estimate ${\kappa }_{1}\left(A\right)$.
${\mathbf{norm}}=\mathrm{Nag_InfNorm}$
Estimate ${\kappa }_{\infty }\left(A\right)$.
Constraint: ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$ or $\mathrm{Nag_InfNorm}$.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     dl[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.
4:     d[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
5:     du[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
6:     du2[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array du2 must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
On entry: must contain the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
7:     ipiv[$\mathit{dim}$]const IntegerInput
Note: the dimension, dim, of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left[i-1\right]$, and ${\mathbf{ipiv}}\left[i-1\right]$ must always be either $i$ or $\left(i+1\right)$, ${\mathbf{ipiv}}\left[i-1\right]=i$ indicating that a row interchange was not performed.
8:     anormdoubleInput
On entry: if ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$, the $1$-norm of the original matrix $A$.
If ${\mathbf{norm}}=\mathrm{Nag_InfNorm}$, the $\infty$-norm of the original matrix $A$.
anorm may be computed as demonstrated in Section 9 for the $1$-norm. The $\infty$-norm may be similarly computed by swapping the dl and du arrays in the code for the $1$-norm.
anorm must be computed either before calling nag_dgttrf (f07cdc) or else from a copy of the original matrix $A$ (see Section 9).
Constraint: ${\mathbf{anorm}}\ge 0.0$.
9:     rconddouble *Output
On exit: contains an estimate of the reciprocal condition number.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{anorm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

7  Accuracy

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The total number of floating point operations required to perform a solve is proportional to $n$.
The complex analogue of this function is nag_zgtcon (f07cuc).

9  Example

This example estimates the condition number in the $1$-norm of the tridiagonal matrix $A$ given by
 $A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 .$

9.1  Program Text

Program Text (f07cgce.c)

9.2  Program Data

Program Data (f07cgce.d)

9.3  Program Results

Program Results (f07cgce.r)