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NAG Toolbox: nag_lapack_dtptri (f07uj)

Purpose

nag_lapack_dtptri (f07uj) computes the inverse of a real triangular matrix, using packed storage.

Syntax

[ap, info] = f07uj(uplo, diag, n, ap)
[ap, info] = nag_lapack_dtptri(uplo, diag, n, ap)

Description

nag_lapack_dtptri (f07uj) forms the inverse of a real triangular matrix AA, using packed storage. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether AA is upper or lower triangular.
uplo = 'U'uplo='U'
AA is upper triangular.
uplo = 'L'uplo='L'
AA is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     diag – string (length ≥ 1)
Indicates whether AA is a nonunit or unit triangular matrix.
diag = 'N'diag='N'
AA is a nonunit triangular matrix.
diag = 'U'diag='U'
AA is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 11.
Constraint: diag = 'N'diag='N' or 'U''U'.
3:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
4:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The nn by nn triangular matrix AA, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of AA must be stored with element AijAij in ap(i + j(j1) / 2)api+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of AA must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.
If diag = 'U'diag='U', the diagonal elements of AA are assumed to be 11, and are not referenced; the same storage scheme is used whether diag = 'N'diag='N' or ‘U’.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
AA stores A1A-1, using the same storage format as described above.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [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 = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: diag, 3: n, 4: ap, 5: info.
W INFO > 0INFO>0
If info = iinfo=i, a(i,i)a(i,i) is exactly zero; AA is singular and its inverse cannot be computed.

Accuracy

The computed inverse XX satisfies
|XAI|c(n)ε|X||A| ,
|XA-I|c(n)ε|X||A| ,
where c(n)c(n) is a modest linear function of nn, and εε is the machine precision.
Note that a similar bound for |AXI||AX-I| cannot be guaranteed, although it is almost always satisfied.
The computed inverse satisfies the forward error bound
|XA1|c(n)ε|A1||A||X| .
|X-A-1|c(n)ε|A-1||A||X| .
See Du Croz and Higham (1992).

Further Comments

The total number of floating point operations is approximately (1/3)n313n3.
The complex analogue of this function is nag_lapack_ztptri (f07uw).

Example

function nag_lapack_dtptri_example
uplo = 'L';
diag = 'N';
n = int64(4);
ap = [4.3;
     -3.96;
     0.4;
     -0.27;
     -4.87;
     0.31;
     0.07;
     -8.02;
     -5.95;
     0.12];
[apOut, info] = nag_lapack_dtptri(uplo, diag, n, ap)
 

apOut =

    0.2326
   -0.1891
    0.0043
    0.8463
   -0.2053
   -0.0079
   -0.2738
   -0.1247
   -6.1825
    8.3333


info =

                    0


function f07uj_example
uplo = 'L';
diag = 'N';
n = int64(4);
ap = [4.3;
     -3.96;
     0.4;
     -0.27;
     -4.87;
     0.31;
     0.07;
     -8.02;
     -5.95;
     0.12];
[apOut, info] = f07uj(uplo, diag, n, ap)
 

apOut =

    0.2326
   -0.1891
    0.0043
    0.8463
   -0.2053
   -0.0079
   -0.2738
   -0.1247
   -6.1825
    8.3333


info =

                    0



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Chapter Introduction
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