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NAG Toolbox: nag_lapack_zpbstf (f08ut)

Purpose

nag_lapack_zpbstf (f08ut) computes a split Cholesky factorization of a complex Hermitian positive definite band matrix.

Syntax

[bb, info] = f08ut(uplo, kb, bb, 'n', n)
[bb, info] = nag_lapack_zpbstf(uplo, kb, bb, 'n', n)

Description

nag_lapack_zpbstf (f08ut) computes a split Cholesky factorization of a complex Hermitian positive definite band matrix BB. It is designed to be used in conjunction with nag_lapack_zhbgst (f08us).
The factorization has the form B = SHSB=SHS, where SS is a band matrix of the same bandwidth as BB and the following structure: SS is upper triangular in the first (n + k) / 2(n+k)/2 rows, and transposed — hence, lower triangular — in the remaining rows. For example, if n = 9n=9 and k = 2k=2, then
S =
  s11 s12 s13 s22 s23 s24 s33 s34 s35 s44 s45 s55 s64 s65 s66 s75 s76 s77 s86 s87 s88 s97 s98 s99  
.
S = s11 s12 s13 s22 s23 s24 s33 s34 s35 s44 s45 s55 s64 s65 s66 s75 s76 s77 s86 s87 s88 s97 s98 s99 .

References

None.

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of BB is stored.
uplo = 'U'uplo='U'
The upper triangular part of BB is stored.
uplo = 'L'uplo='L'
The lower triangular part of BB is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     kb – int64int32nag_int scalar
If uplo = 'U'uplo='U', the number of superdiagonals, kbkb, of the matrix BB.
If uplo = 'L'uplo='L', the number of subdiagonals, kbkb, of the matrix BB.
Constraint: kb0kb0.
3:     bb(ldbb, : :) – complex array
The first dimension of the array bb must be at least kb + 1kb+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 BB.
The matrix is stored in rows 11 to kb + 1kb+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of BB within the band must be stored with element BijBij in bb(kb + 1 + ij,j)​ for ​max (1,jkb)ijbbkb+1+i-jj​ for ​max(1,j-kb)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of BB within the band must be stored with element BijBij in bb(1 + ij,j)​ for ​jimin (n,j + kb).bb1+i-jj​ for ​jimin(n,j+kb).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array bb and the second dimension of the array bb. (An error is raised if these dimensions are not equal.)
nn, the order of the matrix BB.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

ldbb

Output Parameters

1:     bb(ldbb, : :) – complex array
The first dimension of the array bb will be kb + 1kb+1
The second dimension of the array will be max (1,n)max(1,n)
ldbbkb + 1ldbbkb+1.
BB stores the elements of its split Cholesky factor SS.
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: kb, 4: bb, 5: ldbb, 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 factorization could not be completed, because the updated element b(i,i)b(i,i) would be the square root of a negative number. Hence BB is not positive definite. This may indicate an error in forming the matrix BB.

Accuracy

The computed factor SS is the exact factor of a perturbed matrix (B + E)(B+E), where
|E|c(k + 1)ε|SH||S|,
|E|c(k+1)ε|SH||S|,
c(k + 1)c(k+1) is a modest linear function of k + 1k+1, and εε is the machine precision. It follows that |eij|c(k + 1)ε×sqrt((biibjj))|eij|c(k+1)ε(biibjj).

Further Comments

The total number of floating point operations is approximately 4n(k + 1)24n (k+1) 2, assuming nknk.
A call to nag_lapack_zpbstf (f08ut) may be followed by a call to nag_lapack_zhbgst (f08us) to solve the generalized eigenproblem Az = λBzAz=λBz, where AA and BB are banded and BB is positive definite.
The real analogue of this function is nag_lapack_dpbstf (f08uf).

Example

function nag_lapack_zpbstf_example
uplo = 'L';
kb = int64(1);
bb = [complex(9.89),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
      1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
[bbOut, info] = nag_lapack_zpbstf(uplo, kb, bb)
 

bbOut =

   3.1448 + 0.0000i   0.9856 + 0.0000i   0.5362 + 0.0000i   1.4731 + 0.0000i
   0.3434 + 0.5501i  -0.0746 - 0.5408i  -0.2240 - 1.5206i   0.0000 + 0.0000i


info =

                    0


function f08ut_example
uplo = 'L';
kb = int64(1);
bb = [complex(9.89),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
      1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
[bbOut, info] = f08ut(uplo, kb, bb)
 

bbOut =

   3.1448 + 0.0000i   0.9856 + 0.0000i   0.5362 + 0.0000i   1.4731 + 0.0000i
   0.3434 + 0.5501i  -0.0746 - 0.5408i  -0.2240 - 1.5206i   0.0000 + 0.0000i


info =

                    0



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Chapter Introduction
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