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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

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

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

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 $r$ right-hand side matrix $B$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b and the dimension of the array d.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs_p$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$r$ , the number of right-hand sides, 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 $\max (1, n)$

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

The $(n - 1)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $L D L^{H}$ factorization of $A$ . (e can also be regarded as the superdiagonal of the unit 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 $info = 0$ , the $n$ by $r$ solution matrix $X$ .
4: $info$ – 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.

$info > 0$: The leading minor of order $_$ is not positive definite, and the solution has 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. See Section 4.4 of Anderson et al. (1999) for further details.

nag_lapack_zptsvx (f07jp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_posdef_tridiag_solve (f04cg) solves

A x = b

and returns a forward error bound and condition estimate. nag_linsys_complex_posdef_tridiag_solve (f04cg) calls nag_lapack_zptsv (f07jn) to solve the equations.

Further Comments

The number of floating-point operations required for the factorization of

A

is proportional to

n

, and the number of floating-point operations required for the solution of the equations is proportional to

n r

, where

r

is the number of right-hand sides.

The real analogue of this function is nag_lapack_dptsv (f07ja).

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 \\ 93.0 + 62.0 i \\ 78.0 - 80.0 i \\ 14.0 - 27.0 i \end{array}) .

Details of the

L D L^{H}

factorization of

A

are also output.

Open in the MATLAB editor: f07jn_example

function f07jn_example


fprintf('f07jn example results\n\n');

% Hermitian tridiagonal A stored as two diagonals
d = [ 16            41          46            21];
e = [ 16 + 16i      18 - 9i      1 - 4i         ];

%RHS
b = [ 64 + 16i;
      93 + 62i;
      78 - 80i;
      14 - 27i];

%Solve
[df, ef, x, info] = f07jn( ...
                           d, e, b);

disp('Solution');
disp(x);
disp('Diagonal elements of the diagonal matrix D');
disp(df);
disp('Sub-diagonal elements of the Cholesky factor L');
disp(ef);

f07jn example results

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

Diagonal elements of the diagonal matrix D
    16     9     1     4

Sub-diagonal elements of the Cholesky factor L
   1.0000 + 1.0000i   2.0000 - 1.0000i   1.0000 - 4.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox