is a diagonal matrix, with positive diagonal elements. nag_lapack_zpttrs (f07js) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form

U^{H} D U

, where

U

is a unit upper bidiagonal matrix.

References

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

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

Specifies the form of the factorization as follows:

$uplo ='U'$: $A = U^{H} D U$ .
$uplo ='L'$: $A = L D L^{H}$ .

Constraint:

uplo ='U'

'L'

2: $d (:)$ – double array

The dimension of the array d must be at least

\max (1, n)

Must contain the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{H}

U^{H} D U

factorization of

A

3: $e (:)$ – complex array

The dimension of the array e must be at least

\max (1, n - 1)

uplo ='U'

, e must contain the

(n - 1)

superdiagonal elements of the unit upper bidiagonal matrix

U

from the

U^{H} D U

factorization of

A

uplo ='L'

, e must contain the

(n - 1)

subdiagonal elements of the unit lower bidiagonal matrix

L

from the

L D L^{H}

factorization of

A

4: $b (ldb :)$ – 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

r

matrix of right-hand sides

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: $b (ldb :)$ – complex array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

The $n$ by $r$ solution matrix $X$ .
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

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.

Following the use of this function nag_lapack_zptcon (f07ju) can be used to estimate the condition number of

A

and nag_lapack_zptrfs (f07jv) can be used to obtain approximate error bounds.

Further Comments

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

A X = B

is proportional to

n r

The real analogue of this function is nag_lapack_dpttrs (f07je).

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 \\ 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}) .

Open in the MATLAB editor: f07js_example

function f07js_example


fprintf('f07js example results\n\n');

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

% Factorize
[df, ef, info] = f07jr( ...
                        d, e);

% RHS
b = [ 64 + 16i,  -16 - 32i;
      93 + 62i,   61 - 66i;
      78 - 80i,   71 - 74i;
      14 - 27i,   35 + 15i];

% Solve
uplo = 'L';
[x, info] = f07js( ...
                   uplo, df, ef, b);

disp('Solution(s)');
disp(x);

f07js example results

Solution(s)
   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

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

Chapter Contents

Chapter Introduction

NAG Toolbox