1.	If $fact ='N'$ , the matrix $A$ is factorized as $A = L D L^{T}$ , where $L$ is a unit lower bidiagonal matrix and $D$ is diagonal. The factorization can also be regarded as having the form $A = U^{T} D U$ .
2.	If the leading $i$ by $i$ principal minor is not positive definite, 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.

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

Constraint:

fact ='F'

'N'

2: $d (:)$ – double array

The dimension of the array d must be at least

\max (1, n)

The

n

diagonal elements of the tridiagonal matrix

A

3: $e (:)$ – double 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

4: $df (:)$ – double array

The dimension of the array df must be at least

\max (1, n)

fact ='F'

, df must contain the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{T}

factorization of

A

5: $ef (:)$ – double array

The dimension of the array ef must be at least

\max (1, n - 1)

fact ='F'

, ef must contain the

(n - 1)

subdiagonal elements of the unit bidiagonal factor

L

from the

L D L^{T}

factorization of

A

6: $b (ldb :)$ – 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.
$n$ , 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: $df (:)$ – double array: The dimension of the array df will be $\max (1, n)$

If $fact ='N'$ , df contains the $n$ diagonal elements of the diagonal matrix $D$ from the $L D L^{T}$ factorization of $A$ .
2: $ef (:)$ – double array: The dimension of the array ef will be $\max (1, n - 1)$

If $fact ='N'$ , ef contains the $(n - 1)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $L D L^{T}$ factorization of $A$ .
3: $x (ldx :)$ – 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 $X$ .
4: $rcond$ – double scalar: The reciprocal condition number of the matrix $A$ . If rcond is less than the machine precision (in particular, if $rcond = 0.0$ ), the matrix is singular to working precision. This condition is indicated by a return code of $info \geq n + 1$ .
5: $ferr (nrhs_p)$ – double array: The forward error bound for each solution vector ${\hat{x}}_{j}$ (the $j$ th column of the solution matrix $X$ ). If $x_{j}$ is the true solution corresponding to ${\hat{x}}_{j}$ , $ferr (j)$ is an estimated upper bound for the magnitude of the largest element in ( ${\hat{x}}_{j} - x_{j}$ ) divided by the magnitude of the largest element in ${\hat{x}}_{j}$ .
6: $berr (nrhs_p)$ – double array: The component-wise relative backward error of each 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).
7: $info$ – 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.

$info > 0 and info \leq n$: The leading minor of order $_$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed. $rcond = 0.0$ is returned.

W $info = n + 1$: $D$ 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 |R| |R^{T}|, where ​ R = L D^{\frac{1}{2}},

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 10.1 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 number of floating-point operations required for the factorization, and for the estimation of the condition number of

A

is proportional to

n

. The number of floating-point operations required for the solution of the equations, and for the estimation of the forward and backward error is proportional to

n r

, where

r

is the number of right-hand sides.

The condition estimation is based upon Equation (15.11) of Higham (2002). For further details of the error estimation, see Section 4.4 of Anderson et al. (1999).

The complex analogue of this function is nag_lapack_zptsvx (f07jp).

Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite tridiagonal matrix

A = (\begin{matrix} 4.0 & - 2.0 & 0 & 0 & 0 \\ - 2.0 & 10.0 & - 6.0 & 0 & 0 \\ 0 & - 6.0 & 29.0 & 15.0 & 0 \\ 0 & 0 & 15.0 & 25.0 & 8.0 \\ 0 & 0 & 0 & 8.0 & 5.0 \end{matrix})

and

B = (\begin{matrix} 6.0 & 10.0 \\ 9.0 & 4.0 \\ 2.0 & 9.0 \\ 14.0 & 65.0 \\ 7.0 & 23.0 \end{matrix}) .

Error estimates for the solutions and an estimate of the reciprocal of the condition number of

A

are also output.

Open in the MATLAB editor: f07jb_example

function f07jb_example


fprintf('f07jb example results\n\n');

% Symmetric tridiagonal A stored as two diagonals
d = [ 4     10     29     25     5];
e = [-2     -6     15     8       ];

% RHS
b = [ 6, 10;
      9,  4;
      2,  9;
     14, 65;
      7, 23];

% Solve
fact = 'Not factored';
df = d;
ef = e;
[df, ef, x, rcond, ferr, berr, info] = ...
  f07jb( ...
         fact, d, e, df, ef, 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\n',rcond);

f07jb example results

Solution(s)
    2.5000    2.0000
    2.0000   -1.0000
    1.0000   -3.0000
   -1.0000    6.0000
    3.0000   -5.0000

Backward errors (machine-dependent)
   0.0e+00   7.4e-17
Estimated forward error bounds (machine-dependent)
   2.4e-14   4.7e-14

Estimate of reciprocal condition number
   9.5e-03

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

Chapter Contents

Chapter Introduction

NAG Toolbox