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NAG Toolbox: nag_lapack_zpbtrf (f07hr)

Purpose

nag_lapack_zpbtrf (f07hr) computes the Cholesky factorization of a complex Hermitian positive definite band matrix.

Syntax

[ab, info] = f07hr(uplo, kd, ab, 'n', n)
[ab, info] = nag_lapack_zpbtrf(uplo, kd, ab, 'n', n)

Description

nag_lapack_zpbtrf (f07hr) forms the Cholesky factorization of a complex Hermitian positive definite band matrix AA either as A = UHUA=UHU if uplo = 'U'uplo='U' or A = LLHA=LLH if uplo = 'L'uplo='L', where UU (or LL) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as AA.

References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of AA is stored and how AA is to be factorized.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored and AA is factorized as UHUUHU, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored and AA is factorized as LLHLLH, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     kd – int64int32nag_int scalar
kdkd, the number of superdiagonals or subdiagonals of the matrix AA.
Constraint: kd0kd0.
3:     ab(ldab, : :) – complex array
The first dimension of the array ab must be at least kd + 1kd+1
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn Hermitian positive definite band matrix AA.
The matrix is stored in rows 11 to kd + 1kd+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of AA within the band must be stored with element AijAij in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ijabkd+1+i-jj​ for ​max(1,j-kd)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of AA within the band must be stored with element AijAij in ab(1 + ij,j)​ for ​jimin (n,j + kd).ab1+i-jj​ for ​jimin(n,j+kd).

Optional Input Parameters

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

Input Parameters Omitted from the MATLAB Interface

ldab

Output Parameters

1:     ab(ldab, : :) – complex array
The first dimension of the array ab will be kd + 1kd+1
The second dimension of the array will be max (1,n)max(1,n)
ldabkd + 1ldabkd+1.
The upper or lower triangle of AA stores the Cholesky factor UU or LL as specified by uplo, 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

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: kd, 4: ab, 5: ldab, 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.
  INFO > 0INFO>0
If info = iinfo=i, the leading minor of order ii is not positive definite and the factorization could not be completed. Hence AA itself is not positive definite. This may indicate an error in forming the matrix AA. There is no function specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_zgbtrf (f07br) or as a full Hermitian matrix, by calling nag_lapack_zhetrf (f07mr).

Accuracy

If uplo = 'U'uplo='U', the computed factor UU is the exact factor of a perturbed matrix A + EA+E, where
|E|c(k + 1)ε|UH||U| ,
|E|c(k+1)ε|UH||U| ,
c(k + 1)c(k+1) is a modest linear function of k + 1k+1, and εε is the machine precision.
If uplo = 'L'uplo='L', a similar statement holds for the computed factor LL. It follows that |eij|c(k + 1)ε×sqrt(aiiajj)|eij|c(k+1)εaiiajj.

Further Comments

The total number of real floating point operations is approximately 4n(k + 1)24n (k+1) 2, assuming nknk.
A call to nag_lapack_zpbtrf (f07hr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpbtrf (f07hd).

Example

function nag_lapack_zpbtrf_example
uplo = 'L';
kd = int64(1);
ab = [complex(9.39),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
      1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
[abOut, info] = nag_lapack_zpbtrf(uplo, kd, ab)
 

abOut =

   3.0643 + 0.0000i   1.1167 + 0.0000i   1.6066 + 0.0000i   0.4289 + 0.0000i
   0.3524 + 0.5646i  -0.0358 - 0.2597i  -0.2054 - 1.3942i   0.0000 + 0.0000i


info =

                    0


function f07hr_example
uplo = 'L';
kd = int64(1);
ab = [complex(9.39),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
      1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
[abOut, info] = f07hr(uplo, kd, ab)
 

abOut =

   3.0643 + 0.0000i   1.1167 + 0.0000i   1.6066 + 0.0000i   0.4289 + 0.0000i
   0.3524 + 0.5646i  -0.0358 - 0.2597i  -0.2054 - 1.3942i   0.0000 + 0.0000i


info =

                    0



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Chapter Introduction
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