nag_lapack_dpftrf (f07wd) computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format.

Syntax

[ar, info] = f07wd(transr, uplo, n, ar)

[ar, info] = nag_lapack_dpftrf(transr, uplo, n, ar)

Description

nag_lapack_dpftrf (f07wd) forms the Cholesky factorization of a real symmetric positive definite matrix

A

either as

A = U^{T} U

uplo ='U'

A = L L^{T}

uplo ='L'

, where

U

is an upper triangular matrix and

L

is a lower triangular, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.

References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn14.pdf

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 whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: The upper triangular part of $A$ is stored, and $A$ is factorized as $U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ is stored, and $A$ is factorized as $L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

3: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

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

The upper or lower triangular part (as specified by uplo) of the

n

n

symmetric matrix

A

, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.

Optional Input Parameters

None.

Output Parameters

1: $ar (n \times (n + 1) / 2)$ – double array: If $info = 0$ , the factor $U$ or $L$ from the Cholesky factorization $A = U^{T} U$ or $A = L L^{T}$ , in the same storage format as $A$ .
2: $info$ – 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 $info > 0$: The leading minor of order $_$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$ . There is no function specifically designed to factorize a symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling nag_lapack_dsytrf (f07md).

Accuracy

uplo ='U'

, the computed factor

U

is the exact factor of a perturbed matrix

A + E

, where

|E| \leq c (n) ε |U^{T}| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

uplo ='L'

, a similar statement holds for the computed factor

L

. It follows that

|e_{i j}| \leq c (n) ε \sqrt{a_{i i} a_{j j}}

Further Comments

The total number of floating-point operations is approximately

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

A call to nag_lapack_dpftrf (f07wd) may be followed by calls to the functions:

nag_lapack_dpftrs (f07we) to solve $A X = B$ ;
nag_lapack_dpftri (f07wj) to compute the inverse of $A$ .

The complex analogue of this function is nag_lapack_zpftrf (f07wr).

Example

This example computes the Cholesky factorization of the matrix

A

, 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 is stored using RFP format.

Open in the MATLAB editor: f07wd_example

function f07wd_example


fprintf('f07wd 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]);

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

if info == 0
  % Convert factor to full array form, and print it
  [a, info] = f01vg(transr, uplo, n, ar);
  fprintf('\n');
  [ifail] = x04ca(uplo, 'n', a, 'Factor');
else
  fprintf('\na is not positive definite.\n');
end

f07wd example results


 Factor
             1          2          3          4
 1      2.0396
 2     -1.5297     1.6401
 3      0.2746    -0.2500     0.7887
 4     -0.0490     0.6737     0.6617     0.5347

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

Chapter Contents

Chapter Introduction

NAG Toolbox