1.	If $fact ='N'$ , the $L U$ decomposition is used to factor the matrix $A$ as $A = L U$ , where $L$ is a product of permutation and unit lower bidiagonal matrices and $U$ is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
2.	If some $u_{i i} = 0$ , so that $U$ is exactly singular, then the function returns with $info = i$ . Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$ . If the reciprocal of the condition number is less than machine precision, $info \geq n + 1$ is returned as a warning, but the function still goes on to solve for $X$ and compute error bounds as described below.
3.	The system of equations is solved for $X$ using the factored form of $A$ .
4.	Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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

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

Parameters

Compulsory Input Parameters

1: $fact$ – string (length ≥ 1)

Specifies whether or not the factorized form of the matrix

A

has been supplied.

$fact ='F'$: dlf, df, duf, du2 and ipiv contain the factorized form of the matrix $A$ . dlf, df, duf, du2 and ipiv will not be modified.
$fact ='N'$: The matrix $A$ will be copied to dlf, df and duf and factorized.

Constraint:

fact ='F'

'N'

2: $trans$ – string (length ≥ 1)

Specifies the form of the system of equations.

$trans ='N'$: $A X = B$ (No transpose).
$trans ='T'$ or $'C'$: $A^{T} X = B$ (Transpose).

Constraint:

trans ='N'

'T'

'C'

3: $dl (:)$ – double array

The dimension of the array dl must be at least

\max (1, n - 1)

The

(n - 1)

subdiagonal elements of

A

4: $d (:)$ – double array

The dimension of the array d must be at least

\max (1, n)

The

n

diagonal elements of

A

5: $du (:)$ – double array

The dimension of the array du must be at least

\max (1, n - 1)

The

(n - 1)

superdiagonal elements of

A

6: $dlf (:)$ – double array

The dimension of the array dlf must be at least

\max (1, n - 1)

fact ='F'

, dlf contains the

(n - 1)

multipliers that define the matrix

L

from the

L U

factorization of

A

7: $df (:)$ – double array

The dimension of the array df must be at least

\max (1, n)

fact ='F'

, df contains the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

8: $duf (:)$ – double array

The dimension of the array duf must be at least

\max (1, n - 1)

fact ='F'

, duf contains the

(n - 1)

elements of the first superdiagonal of

U

9: $du2 (:)$ – double array

The dimension of the array du2 must be at least

\max (1, n - 2)

fact ='F'

, du2 contains the (

n - 2

) elements of the second superdiagonal of

U

10: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

fact ='F'

, ipiv contains the pivot indices from the

L U

factorization of

A

11: $b (ldb :)$ – 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

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

If $fact ='N'$ , dlf contains the $(n - 1)$ multipliers that define the matrix $L$ from the $L U$ factorization of $A$ .
2: $df (:)$ – double array: The dimension of the array df will be $\max (1, n)$

If $fact ='N'$ , df contains the $n$ diagonal elements of the upper triangular matrix $U$ from the $L U$ factorization of $A$ .
3: $duf (:)$ – double array: The dimension of the array duf will be $\max (1, n - 1)$

If $fact ='N'$ , duf contains the $(n - 1)$ elements of the first superdiagonal of $U$ .
4: $du2 (:)$ – double array: The dimension of the array du2 will be $\max (1, n - 2)$

If $fact ='N'$ , du2 contains the ( $n - 2$ ) elements of the second superdiagonal of $U$ .
5: $ipiv (:)$ – int64int32nag_int array: The dimension of the array ipiv will be $\max (1, n)$

If $fact ='N'$ , ipiv contains the pivot indices from the $L U$ factorization of $A$ ; row $i$ of the matrix was interchanged with row $ipiv (i)$ . $ipiv (i)$ will always be either $i$ or $i + 1$ ; $ipiv (i) = i$ indicates a row interchange was not required.
6: $x (ldx :)$ – double array: The first dimension of the array x will be $\max (1, n)$ .
The second dimension of the array x will be $\max (1, nrhs_p)$ .

If $info = 0$ or $n + 1$ , the $n$ by $r$ solution matrix $X$ .
7: $rcond$ – double scalar: The estimate of the reciprocal condition number of the matrix $A$ . If $rcond = 0.0$ , the matrix may be exactly singular. This condition is indicated by $info > 0 and info \leq n$ . Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by $info \geq n + 1$ .
8: $ferr (nrhs_p)$ – double array: If $info = 0$ or $n + 1$ , an estimate of the forward error bound for each computed solution vector, such that ${‖{\hat{x}}_{j} - x_{j}‖}_{\infty} / {‖x_{j}‖}_{\infty} \leq ferr (j)$ where ${\hat{x}}_{j}$ is the $j$ th column of the computed solution returned in the array x and $x_{j}$ is the corresponding column of the exact solution $X$ . The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
9: $berr (nrhs_p)$ – double array: If $info = 0$ or $n + 1$ , an estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).
10: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

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

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W $info > 0 and info < n$: Element $_$ of the diagonal is exactly zero. The factorization has not been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond = 0.0$ is returned.

W $info > 0 and info = n$: Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond = 0.0$ is returned.

W $info = n + 1$: $U$ is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

Accuracy

For each right-hand side vector

b

, the computed solution

\hat{x}

is the exact solution of a perturbed system of equations

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

, where

|E| \leq c (n) ε |L| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 9.3 of Higham (2002) for further details.

x

is the true solution, then the computed solution

\hat{x}

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖\hat{x}‖}_{\infty}} \leq w_{c} cond (A, \hat{x}, b)

where

cond (A, \hat{x}, b) = {‖|A^{- 1}| (|A| |\hat{x}| + |b|)‖}_{\infty} / {‖\hat{x}‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

. If

\hat{x}

is the

j

th column of

X

, then

w_{c}

is returned in

berr (j)

and a bound on

{‖x - \hat{x}‖}_{\infty} / {‖\hat{x}‖}_{\infty}

is returned in

ferr (j)

. 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 typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of this function is nag_lapack_zgtsvx (f07cp).

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}) .

Estimates for the backward errors, forward errors and condition number are also output.

Open in the MATLAB editor: f07cb_example

function f07cb_example


fprintf('f07cb 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);

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

% Initialize input parameters
fact  = 'No factors';
trans = 'No transpose';
dlf = dl;
df  = d;
duf = du;
du2 = zeros(n-2, 1);
ipiv = zeros(n,1,'int64');

% Solve AX = B
[dlf, df, duf, du2, ipiv, x, rcond, ferr, berr, info] = ...
  f07cb( ...
         fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b);

disp('Solution(s)');
disp(x);
disp('Backward errors (machine-dependent)');
fprintf('%10.1e',berr);
fprintf('\n');
disp('Estimated forward error bounds (machine-dependent)');
fprintf('%10.1e',ferr);
fprintf('\n\n');
disp('Estimate of reciprocal condition number');
fprintf('%10.1e\n',rcond);

f07cb example results

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

Backward errors (machine-dependent)
   7.2e-17   5.9e-17
Estimated forward error bounds (machine-dependent)
   9.4e-15   1.4e-14

Estimate of reciprocal condition number
   1.1e-02

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

Chapter Contents

Chapter Introduction

NAG Toolbox