Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

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

Parameters

Compulsory Input Parameters

1: $d (:)$ – double array: The dimension of the array d must be at least $\max (1, n)$

Must contain the $n$ diagonal elements of the tridiagonal matrix $A$ .
2: $e (:)$ – complex array: The dimension of the array e must be at least $\max (1, n - 1)$

Must contain the $(n - 1)$ subdiagonal elements of the tridiagonal matrix $A$ .
3: $b (ldb :)$ – complex array: The first dimension of the array b must be at least $\max (1, n)$ .
The second dimension of the array b must be at least $\max (1, nrhs_p)$ .

The $n$ by $r$ matrix of right-hand sides $B$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b.
The number of linear equations $n$ , i.e., the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs_p$ – int64int32nag_int scalar: Default: the second dimension of the array b.
The number of right-hand sides $r$ , i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $d (:)$ – double array: The dimension of the array d will be $\max (1, n)$

If $ifail = 0$ or $n + 1$ , d stores the $n$ diagonal elements of the diagonal matrix $D$ from the $L D L^{H}$ factorization of $A$ .
2: $e (:)$ – complex array: The dimension of the array e will be $\max (1, n - 1)$

If $ifail = 0$ or $n + 1$ , e stores the $(n - 1)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ from the $L D L^{H}$ factorization of $A$ . (e can also be regarded as the conjugate of the superdiagonal of the unit upper bidiagonal factor $U$ from the $U^{H} D U$ factorization of $A$ .)
3: $b (ldb :)$ – complex array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

If $ifail = 0$ or $n + 1$ , the $n$ by $r$ solution matrix $X$ .
4: $rcond$ – double scalar: If $ifail = 0$ or $n + 1$ , an estimate of the reciprocal of the condition number of the matrix $A$ , computed as $rcond = 1 / ({‖A‖}_{1}, {‖A^{- 1}‖}_{1})$ .
5: $errbnd$ – double scalar: If $ifail = 0$ or $n + 1$ , an estimate of the forward error bound for a computed solution $\hat{x}$ , such that ${‖\hat{x} - x‖}_{1} / {‖x‖}_{1} \leq errbnd$ , where $\hat{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$ . If rcond is less than machine precision, then errbnd is returned as unity.
6: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail > 0 and ifail \leq n$: The principal minor of order $_$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.

W $ifail = n + 1$: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

$ifail = - 1$: Constraint: $n \geq 0$ .

$ifail = - 2$: Constraint: $nrhs_p \geq 0$ .

$ifail = - 6$: Constraint: $ldb \geq \max (1, n)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.
The double allocatable memory required is n. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.

Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. nag_linsys_complex_posdef_tridiag_solve (f04cg) uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The total number of floating-point operations required to solve the equations

A X = B

is proportional to

n r

. The condition number estimation requires

O (n)

floating-point operations.

See Section 15.3 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.

The real analogue of nag_linsys_complex_posdef_tridiag_solve (f04cg) is nag_linsys_real_posdef_tridiag_solve (f04bg).

Example

This example solves the equations

A X = B,

where

A

is the Hermitian positive definite tridiagonal matrix

A = (\begin{array}{r} 16.0 & 16.0 + 16.0 i & 0 & 0 \\ 16.0 - 16.0 i & 41.0 & 18.0 - 9.0 i & 0 \\ 0 & 18.0 + 9.0 i & 46.0 & 1.0 - 4.0 i \\ 0 & 0 & 1.0 + 4.0 i & 21.0 \end{array})

and

B = (\begin{array}{r} 64.0 + 16.0 i & - 16.0 - 32.0 i \\ 93.0 + 62.0 i & 61.0 - 66.0 i \\ 78.0 - 80.0 i & 71.0 - 74.0 i \\ 14.0 - 27.0 i & 35.0 + 15.0 i \end{array}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

Open in the MATLAB editor: f04cg_example

function f04cg_example


fprintf('f04cg example results\n\n');

% Solve complex Ax = b for Hermitian tridiagonal A 
% with error bound and condition number.
d = [ 16         41         46        21];
e = [ 16 + 16i   18 -  9i    1 - 4i     ];
b = [ 64 + 16i, -16 - 32i;
      93 + 62i,  61 - 66i;
      78 - 80i,  71 - 74i;
      14 - 27i,  35 + 15i];

[d, e, x, rcond, errbnd, ifail] = ...
  f04cg(d, e, b);

disp('Solution');
disp(x);
disp('Estimate of condition number');
fprintf('%10.1f\n\n',1/rcond);
disp('Estimate of error bound for computed solutions');
fprintf('%10.1e\n\n',errbnd);

f04cg example results

Solution
   2.0000 + 1.0000i  -3.0000 - 2.0000i
   1.0000 + 1.0000i   1.0000 + 1.0000i
   1.0000 - 2.0000i   1.0000 - 2.0000i
   1.0000 - 1.0000i   2.0000 + 1.0000i

Estimate of condition number
    9206.6

Estimate of error bound for computed solutions
   1.0e-12

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

Chapter Contents

Chapter Introduction

NAG Toolbox