nag_lapack_zgttrs (f07cs) 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_zgttrs (f07cs) then utilizes the factorization to solve the required equations.

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

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

Parameters

Compulsory Input Parameters

1: $trans$ – string (length ≥ 1)

Specifies the equations to be solved as follows:

$trans ='N'$: Solve $A X = B$ for $X$ .
$trans ='T'$: Solve $A^{T} X = B$ for $X$ .
$trans ='C'$: Solve $A^{H} X = B$ for $X$ .

Constraint:

trans ='N'

'T'

'C'

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: $b (ldb :)$ – 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 arrays d, ipiv.
$n$ , 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_zgtcon (f07cu) can be used to estimate the condition number of

A

and nag_lapack_zgtrfs (f07cv) 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

A^{T} X = B

A^{H} X = B

is proportional to

n r

The real analogue of this function is nag_lapack_dgttrs (f07ce).

Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

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})

and

B = (\begin{array}{r} 2.4 - 5.0 i & 2.7 + 6.9 i \\ 3.4 + 18.2 i & - 6.9 - 5.3 i \\ - 14.7 + 9.7 i & - 6.0 - 0.6 i \\ 31.9 - 7.7 i & - 3.9 + 9.3 i \\ - 1.0 + 1.6 i & - 3.0 + 12.2 i \end{array}) .

Open in the MATLAB editor: f07cs_example

function f07cs_example


fprintf('f07cs 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               ];

% Rhs B
b = [  2.4 -  5.0i   2.7 +  6.9i; 
       3.4 + 18.2i  -6.9 -  5.3i; 
     -14.7 +  9.7i  -6.0 -  0.6i; 
      31.9 -  7.7i  -3.9 +  9.3i; 
      -1   +  1.6i  -3.0 + 12.2i];

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

% Solve
trans = 'No transpose';
[x, info] = f07cs( ...
                   trans, dlf, df, duf, du2, ipiv, b);

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

f07cs example results

Solution(s)
   1.0000 + 1.0000i   2.0000 - 1.0000i
   3.0000 - 1.0000i   1.0000 + 2.0000i
   4.0000 + 5.0000i  -1.0000 + 1.0000i
  -1.0000 - 2.0000i   2.0000 + 1.0000i
   1.0000 - 1.0000i   2.0000 - 2.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox