nag_lapack_zgtcon (f07cu) should be preceded by a call to nag_lapack_zgttrf (f07cr), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

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_lapack_zgtcon (f07cu) then utilizes the factorization to estimate either

{‖A^{- 1}‖}_{1}

{‖A^{- 1}‖}_{\infty}

, from which the estimate of the reciprocal of the condition number of

A

1 / κ (A)

is computed as either

1 / κ_{1} (A) = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

1 / κ_{\infty} (A) = 1 / ({‖A‖}_{\infty} {‖A^{- 1}‖}_{\infty}) .

1 / κ (A)

is returned, rather than

κ (A)

, since when

A

is singular

κ (A)

is infinite.

Note that

κ_{\infty} (A) = κ_{1} (A^{T})

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)

$norm_p ='1'$ or $'O'$: Estimate $κ_{1} (A)$ .
$norm_p ='I'$: Estimate $κ_{\infty} (A)$ .

Constraint:

norm_p ='1'

'O'

'I'

2: $dl (:)$ – complex array

The dimension of the array dl must be at least

\max (1, n - 1)

Must contain the

(n - 1)

multipliers that define the matrix

L

of the

L U

factorization of

A

3: $d (:)$ – complex array

The dimension of the array d must be at least

\max (1, n)

Must contain the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

4: $du (:)$ – complex array

The dimension of the array du must be at least

\max (1, n - 1)

Must contain the

(n - 1)

elements of the first superdiagonal of

U

5: $du2 (:)$ – complex array

The dimension of the array du2 must be at least

\max (1, n - 2)

Must contain the

(n - 2)

elements of the second superdiagonal of

U

6: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

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

ipiv (i)

, and

ipiv (i)

must always be either

i

(i + 1)

ipiv (i) = i

indicating that a row interchange was not performed.

7: $anorm$ – double scalar

norm_p ='1'

'O'

, the

1

-norm of the original matrix

A

norm_p ='I'

, the

\infty

-norm of the original matrix

A

anorm must be computed either before calling nag_lapack_zgttrf (f07cr) or else from a copy of the original matrix

A

(see Example).

Constraint:

anorm \geq 0.0

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the arrays d, ipiv.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

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

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

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.

Further Comments

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 real analogue of this function is nag_lapack_dgtcon (f07cg).

Example

This example estimates the condition number in the

1

-norm of the tridiagonal matrix

A

given by

A = (\begin{array}{r} - 1.3 + 1.3 i & 2.0 - 1.0 i & 0 & 0 & 0 \\ 1.0 - 2.0 i & - 1.3 + 1.3 i & 2.0 + 1.0 i & 0 & 0 \\ 0 & 1.0 + 1.0 i & - 1.3 + 3.3 i & - 1.0 + 1.0 i & 0 \\ 0 & 0 & 2.0 - 3.0 i & - 0.3 + 4.3 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 1.0 + 1.0 i & - 3.3 + 1.3 i \end{array}) .

Open in the MATLAB editor: f07cu_example

function f07cu_example


fprintf('f07cu example results\n\n');

% Tridiagonal matrix stored by diagonals
du = [              2   - 1i     2   + 1i    -1   + 1i     1   - 1i  ];
d  = [-1.3 + 1.3i  -1.3 + 1.3i  -1.3 + 3.3i  -0.3 + 4.3i  -3.3 + 1.3i];
dl = [ 1   - 2i     1   + 1i     2   - 3i     1   + 1i               ];

% Factorize.
[dlf, df, duf, du2, ipiv, info] = ...
  f07cr(dl, d, du);

% 1-norm of A (construct column preserving matrix)
a = [0+0i du; d; dl 0+0i];
anorm = norm(a,1);

% Estimate condition number 
norm_p = '1-norm';
[rcond, info] = f07cu( ...
                       norm_p, dlf, df, duf, du2, ipiv, anorm);

fprintf('Estimate of condition number = %10.2e\n', 1/rcond);

f07cu example results

Estimate of condition number =   1.84e+02

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox