1.	If $fact ='N'$ , the diagonal pivoting method is used to factor $A$ as $A = U D U^{T}$ if $uplo ='U'$ or $A = L D L^{T}$ if $uplo ='L'$ , where $U$ (or $L$ ) is a product of permutation and unit upper (lower) triangular matrices and $D$ is symmetric and block diagonal with $1$ by $1$ and $2$ by $2$ diagonal blocks.
2.	If some $d_{i i} = 0$ , so that $D$ 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.

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

Constraint:

fact ='F'

'N'

2: $uplo$ – string (length ≥ 1)

uplo ='U'

, the upper triangle of

A

is stored.

uplo ='L'

, the lower triangle of

A

is stored.

Constraint:

uplo ='U'

'L'

3: $ap (:)$ – double array

The dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

The

n

n

symmetric matrix

A

, packed by columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

4: $afp (:)$ – double array

The dimension of the array afp must be at least

\max (1, n \times (n + 1) / 2)

fact ='F'

, afp contains the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

from the factorization

A = U D U^{T}

A = L D L^{T}

as computed by nag_lapack_dsptrf (f07pd), stored as a packed triangular matrix in the same storage format as

A

5: $ipiv (n)$ – int64int32nag_int array

fact ='F'

, ipiv contains details of the interchanges and the block structure of

D

, as determined by nag_lapack_dsptrf (f07pd).

if $ipiv (i) = k > 0$ , $d_{i i}$ is a $1$ by $1$ pivot block and the $i$ th row and column of $A$ were interchanged with the $k$ th row and column;
if $uplo ='U'$ and $ipiv (i - 1) = ipiv (i) = - l < 0$ , $(\begin{matrix} d_{i - 1, i - 1} & {\bar{d}}_{i, i - 1} \\ {\bar{d}}_{i, i - 1} & d_{i i} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i - 1)$ th row and column of $A$ were interchanged with the $l$ th row and column;
if $uplo ='L'$ and $ipiv (i) = ipiv (i + 1) = - m < 0$ , $(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i + 1)$ th row and column of $A$ were interchanged with the $m$ th row and column.

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 dimension of the array ipiv.
$n$ , the number of linear equations, i.e., 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: $afp (:)$ – double array: The dimension of the array afp will be $\max (1, n \times (n + 1) / 2)$

If $fact ='N'$ , afp contains the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A = U D U^{T}$ or $A = L D L^{T}$ as computed by nag_lapack_dsptrf (f07pd), stored as a packed triangular matrix in the same storage format as $A$ .
2: $ipiv (n)$ – int64int32nag_int array: If $fact ='N'$ , ipiv contains details of the interchanges and the block structure of $D$ , as determined by nag_lapack_dsptrf (f07pd), as described above.
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 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 $info \geq n + 1$ .
5: $ferr (nrhs_p)$ – 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.
6: $berr (nrhs_p)$ – 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).
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.

W $info > 0 and info \leq n$: Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the factor $D$ is exactly singular, so the solution and error bounds could not be 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‖}_{1} = O (ε) {‖A‖}_{1},

where

ε

is the machine precision. See Chapter 11 of Higham (2002) for further details.

\hat{x}

is the true solution, then the computed solution

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 factorization of

A

requires approximately

\frac{1}{3} n^{3}

floating-point operations.

For each right-hand side, computation of the backward error involves a minimum of

4 n^{2}

floating-point operations. Each step of iterative refinement involves an additional

6 n^{2}

operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form

A x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n^{2}

operations.

The complex analogues of this function are nag_lapack_zhpsvx (f07pp) for Hermitian matrices, and nag_lapack_zspsvx (f07qp) for symmetric matrices.

Example

This example solves the equations

A X = B,

where

A

is the symmetric matrix

A = (\begin{matrix} - 1.81 & 2.06 & 0.63 & - 1.15 \\ 2.06 & 1.15 & 1.87 & 4.20 \\ 0.63 & 1.87 & - 0.21 & 3.87 \\ - 1.15 & 4.20 & 3.87 & 2.07 \end{matrix}) and B = (\begin{matrix} 0.96 & 3.93 \\ 6.07 & 19.25 \\ 8.38 & 9.90 \\ 9.50 & 27.85 \end{matrix}) .

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

A

are also output.

Open in the MATLAB editor: f07pb_example

function f07pb_example


fprintf('f07pb example results\n\n');

% Symmetric matrix, upper triangle stored in packed format
uplo = 'U';
ap = [-1.81;    
       2.06;    1.15;
       0.63;    1.87;   -0.21;
      -1.15;    4.2;     3.87;    2.07];
% RHS
b = [0.96,  3.93;
     6.07, 19.25;
     8.38,  9.90;
     9.50, 27.85];

% Factorize and solve
fact = 'Not Factorized';
ipiv = zeros(4,1,'int64');
afp  = ap;

[afp, ipiv, x, rcond, ferr, berr, info] = ...
  f07pb(...
        fact, uplo, ap, afp, ipiv, b);

disp('Solution');
disp(x);
fprintf('Condition number      = %7.3f\n',1/rcond);
fprintf('Forward  error bounds = %10.3e  %10.3e\n',ferr); 
fprintf('Backward error bounds = %10.3e  %10.3e\n',berr);

f07pb example results

Solution
   -5.0000    2.0000
   -2.0000    3.0000
    1.0000    4.0000
    4.0000    1.0000

Condition number      =  75.687
Forward  error bounds =  2.424e-14   3.379e-14
Backward error bounds =  9.514e-17   8.933e-17

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

Chapter Contents

Chapter Introduction

NAG Toolbox