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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgtcon (f07cg)

## Purpose

nag_lapack_dgtcon (f07cg) estimates the reciprocal condition number of a real n $n$ by n $n$ tridiagonal matrix A $A$, using the LU $LU$ factorization returned by nag_lapack_dgttrf (f07cd).

## Syntax

[rcond, info] = f07cg(norm_p, dl, d, du, du2, ipiv, anorm, 'n', n)
[rcond, info] = nag_lapack_dgtcon(norm_p, dl, d, du, du2, ipiv, anorm, 'n', n)

## Description

nag_lapack_dgtcon (f07cg) should be preceded by a call to nag_lapack_dgttrf (f07cd), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A $A$ as
 A = PLU , $A=PLU ,$
where P $P$ is a permutation matrix, L $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and U $U$ is an upper triangular band matrix, with two superdiagonals. nag_lapack_dgtcon (f07cg) then utilizes the factorization to estimate either A11 ${‖{A}^{-1}‖}_{1}$ or A1 ${‖{A}^{-1}‖}_{\infty }$, from which the estimate of the reciprocal of the condition number of A $A$, 1 / κ(A) $1/\kappa \left(A\right)$ is computed as either
 1 / κ1 (A) = 1 / (‖A‖1‖A − 1‖1) $1 / κ1 (A) = 1 / ( ‖A‖1 ‖A-1‖1 )$
or
 1 / κ∞ (A) = 1 / (‖A‖∞‖A − 1‖∞) . $1 / κ∞ (A) = 1 / ( ‖A‖∞ ‖A-1‖∞ ) .$
1 / κ(A) $1/\kappa \left(A\right)$ is returned, rather than κ(A) $\kappa \left(A\right)$, since when A $A$ is singular κ(A) $\kappa \left(A\right)$ is infinite.
Note that κ(A) = κ1(AT) ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     norm_p – string (length ≥ 1)
Specifies the norm to be used to estimate κ(A)$\kappa \left(A\right)$.
norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$
Estimate κ1(A)${\kappa }_{1}\left(A\right)$.
norm = 'I'${\mathbf{norm}}=\text{'I'}$
Estimate κ(A)${\kappa }_{\infty }\left(A\right)$.
Constraint: norm = '1'${\mathbf{norm}}=\text{'1'}$, 'O'$\text{'O'}$ or 'I'$\text{'I'}$.
2:     dl( : $:$) – double array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ multipliers that define the matrix L$L$ of the LU$LU$ factorization of A$A$.
3:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ diagonal elements of the upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
4:     du( : $:$) – double array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
5:     du2( : $:$) – double array
Note: the dimension of the array du2 must be at least max (1,n2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
Must contain the (n2)$\left(n-2\right)$ elements of the second superdiagonal of U$U$.
6:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ pivot indices that define the permutation matrix P$P$. At the i$i$th step, row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$, and ipiv(i)${\mathbf{ipiv}}\left(i\right)$ must always be either i$i$ or (i + 1)$\left(i+1\right)$, ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
7:     anorm – double scalar
If norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$, the 1$1$-norm of the original matrix A$A$.
If norm = 'I'${\mathbf{norm}}=\text{'I'}$, the $\infty$-norm of the original matrix A$A$.
anorm may be computed by calling nag_blas_dlangt (f06rn) with the same value for the parameter norm_p.
anorm must be computed either before calling nag_lapack_dgttrf (f07cd) or else from a copy of the original matrix A$A$ (see Section [Example]).
Constraint: anorm0.0${\mathbf{anorm}}\ge 0.0$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the arrays d, ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

work iwork

### Output Parameters

1:     rcond – double scalar
Contains an estimate of the reciprocal condition number.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: norm_p, 2: n, 3: dl, 4: d, 5: du, 6: du2, 7: ipiv, 8: anorm, 9: rcond, 10: work, 11: iwork, 12: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## 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 $n$.
The complex analogue of this function is nag_lapack_zgtcon (f07cu).

## Example

```function nag_lapack_dgtcon_example
norm_p = '1-norm';
dl = [0.8823529411764706;
0.01960784313725495;
0.1400560224089636;
-0.01479925303454714];
d = [3.4;
3.6;
7;
-6;
-1.015373482726424];
du = [2.3;
-5;
-0.9;
7.1];
du2 = [-1;
1.9;
8];
ipiv = [int64(2);3;4;5;5];
anorm = 15.1;
[rcond, info] = nag_lapack_dgtcon(norm_p, dl, d, du, du2, ipiv, anorm)
```
```

rcond =

0.0108

info =

0

```
```function f07cg_example
norm_p = '1-norm';
dl = [0.8823529411764706;
0.01960784313725495;
0.1400560224089636;
-0.01479925303454714];
d = [3.4;
3.6;
7;
-6;
-1.015373482726424];
du = [2.3;
-5;
-0.9;
7.1];
du2 = [-1;
1.9;
8];
ipiv = [int64(2);3;4;5;5];
anorm = 15.1;
[rcond, info] = f07cg(norm_p, dl, d, du, du2, ipiv, anorm)
```
```

rcond =

0.0108

info =

0

```