Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

Parameters

Compulsory Input Parameters

1: $transr$ – string (length ≥ 1)

Specifies whether the RFP representation of

A

is normal or transposed.

$transr ='N'$: The matrix $A$ is stored in normal RFP format.
$transr ='T'$: The matrix $A$ is stored in transposed RFP format.

Constraint:

transr ='N'

'T'

2: $uplo$ – string (length ≥ 1)

Specifies how

A

has been factorized.

$uplo ='U'$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

3: $ar (n \times (n + 1) / 2)$ – double array

The Cholesky factorization of

A

stored in RFP format, as returned by nag_lapack_dpftrf (f07wd).

4: $b (ldb :)$ – 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 ar.
$n$ , 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.

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

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

if $uplo ='U'$ , $|E| \leq c (n) ε |U^{T}| |U|$ ;
if $uplo ='L'$ , $|E| \leq c (n) ε |L| |L^{T}|$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

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

where

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

and

κ_{\infty} (A)

is the condition number when using the

\infty

-norm.

Note that

cond (A, x)

can be much smaller than

cond (A)

Further Comments

The total number of floating-point operations is approximately

2 n^{2} r

The complex analogue of this function is nag_lapack_zpftrs (f07ws).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) and B = (\begin{matrix} 8.70 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \end{matrix}) .

Here

A

is symmetric positive definite, stored in RFP format, and must first be factorized by nag_lapack_dpftrf (f07wd).

Open in the MATLAB editor: f07we_example

function f07we_example


fprintf('f07we example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 0.76   0.34; 
       4.16   1.18;
      -3.12   5.03;
       0.56  -0.83;
      -0.10   1.18];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% RHS
b = [  8.70,  8.30;
     -13.35,  2.13;
       1.89,  1.61;
      -4.14,  5.00];

% Factorize a
[ar, info] = f07wd(transr, uplo, n, ar);

if info == 0
  % Compute solution
  [b, info] = f07we(transr, uplo, ar, b);
  fprintf('\n');
  [ifail] = x04ca('g', ' ', b, 'Solutions');
else
  fprintf('\na is not positive definite.\n');
end

f07we example results


 Solutions
             1          2
 1      1.0000     4.0000
 2     -1.0000     3.0000
 3      2.0000     2.0000
 4     -3.0000     1.0000

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

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