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NAG Toolbox: nag_lapack_ztrtri (f07tw)

Purpose

nag_lapack_ztrtri (f07tw) computes the inverse of a complex triangular matrix.

Syntax

[a, info] = f07tw(uplo, diag, a, 'n', n)
[a, info] = nag_lapack_ztrtri(uplo, diag, a, 'n', n)

Description

nag_lapack_ztrtri (f07tw) forms the inverse of a complex triangular matrix AA. 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:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn triangular matrix AA.
  • If uplo = 'U'uplo='U', aa is upper triangular and the elements of the array below the diagonal are not referenced.
  • If uplo = 'L'uplo='L', aa is lower triangular and the elements of the array above the diagonal are not referenced.
  • If diag = 'U'diag='U', the diagonal elements of aa are assumed to be 11, and are not referenced.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
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: a, 5: lda, 6: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
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 real floating point operations is approximately (4/3)n343n3.
The real analogue of this function is nag_lapack_dtrtri (f07tj).

Example

function nag_lapack_ztrtri_example
uplo = 'L';
diag = 'N';
a = [ 4.78 + 4.56i,  0 + 0i,  0 + 0i,  0 + 0i;
      2 - 0.3i,  -4.11 + 1.25i,  0 + 0i,  0 + 0i;
      2.89 - 1.34i,  2.36 - 4.25i,  4.15 + 0.8i,  0 + 0i;
      -1.89 + 1.15i,  0.04 - 3.69i,  -0.02 + 0.46i,  0.33 - 0.26i];
[aOut, info] = nag_lapack_ztrtri(uplo, diag, a)
 

aOut =

   0.1095 - 0.1045i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0582 - 0.0411i  -0.2227 - 0.0677i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0032 + 0.1905i   0.1538 - 0.2192i   0.2323 - 0.0448i   0.0000 + 0.0000i
   0.7602 + 0.2814i   1.6184 - 1.4346i   0.1289 - 0.2250i   1.8697 + 1.4731i


info =

                    0


function f07tw_example
uplo = 'L';
diag = 'N';
a = [ 4.78 + 4.56i,  0 + 0i,  0 + 0i,  0 + 0i;
      2 - 0.3i,  -4.11 + 1.25i,  0 + 0i,  0 + 0i;
      2.89 - 1.34i,  2.36 - 4.25i,  4.15 + 0.8i,  0 + 0i;
      -1.89 + 1.15i,  0.04 - 3.69i,  -0.02 + 0.46i,  0.33 - 0.26i];
[aOut, info] = f07tw(uplo, diag, a)
 

aOut =

   0.1095 - 0.1045i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0582 - 0.0411i  -0.2227 - 0.0677i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0032 + 0.1905i   0.1538 - 0.2192i   0.2323 - 0.0448i   0.0000 + 0.0000i
   0.7602 + 0.2814i   1.6184 - 1.4346i   0.1289 - 0.2250i   1.8697 + 1.4731i


info =

                    0



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Chapter Introduction
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