nag_lapack_dgttrs (f07ce) should be preceded by a call to nag_lapack_dgttrf (f07cd), 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_dgttrs (f07ce) 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

Parameters

Compulsory Input Parameters

1: $trans$ – string (length ≥ 1)

Specifies the equations to be solved as follows:

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

Constraint:

trans ='N'

'T'

'C'

2: $dl (:)$ – double 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: $d (:)$ – double 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: $du (:)$ – double 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: $du2 (:)$ – double 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: $b (ldb :)$ – double 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: $b (ldb :)$ – double 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_dgtcon (f07cg) can be used to estimate the condition number of

A

and nag_lapack_dgtrfs (f07ch) 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

is proportional to

n r

The complex analogue of this function is nag_lapack_zgttrs (f07cs).

Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) and B = (\begin{matrix} 2.7 & 6.6 \\ - 0.5 & 10.8 \\ 2.6 & - 3.2 \\ 0.6 & - 11.2 \\ 2.7 & 19.1 \end{matrix}) .

Open in the MATLAB editor: f07ce_example

function f07ce_example


fprintf('f07ce example results\n\n');

% Tridiagonal matrix A stored as diagonals:
du = [        2.1    -1.0      1.9     8.0];
d  = [3.0     2.3    -5.0     -0.9     7.1];
dl = [3.4     3.6     7.0     -6.0        ];
n  = numel(d);

% Factorize A.
[dlf, df, duf, du2f, ipiv, info] = ...
  f07cd(dl, d, du);

% Rhs B
b = [ 2.7,   6.6;
     -0.5,  10.8;
      2.6,  -3.2;
      0.6, -11.2;
      2.7,  19.1];

% Solve AX = B
trans = 'No transpose';
[x, info] = f07ce( ...
                   trans, dlf, df, duf, du2f, ipiv, b);

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

f07ce example results

Solution(s)
   -4.0000    5.0000
    7.0000   -4.0000
    3.0000   -3.0000
   -4.0000   -2.0000
   -3.0000    1.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox