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NAG Toolbox: nag_linsys_complex_posdef_packed_solve (f04ce)

Purpose

nag_linsys_complex_posdef_packed_solve (f04ce) computes the solution to a complex system of linear equations AX = BAX=B, where AA is an nn by nn Hermitian positive definite matrix, stored in packed format, and XX and BB are nn by rr matrices. An estimate of the condition number of AA and an error bound for the computed solution are also returned.

Syntax

[ap, b, rcond, errbnd, ifail] = f04ce(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[ap, b, rcond, errbnd, ifail] = nag_linsys_complex_posdef_packed_solve(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The Cholesky factorization is used to factor AA as A = UHUA=UHU, if uplo = 'U'uplo='U', or A = LLHA=LLH, if uplo = 'L'uplo='L', where UU is an upper triangular matrix and LL is a lower triangular matrix. The factored form of AA is then used to solve the system of equations AX = BAX=B.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangle of the matrix AA is stored.
If uplo = 'L'uplo='L', the lower triangle of the matrix AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     ap( : :) – complex 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 Hermitian matrix AA. The upper or lower triangular part of the Hermitian matrix is packed column-wise in a linear array. The jjth column of AA is stored in the array ap as follows:
  • if uplo = 'U'uplo='U', ap(i + (j1)j / 2) = aijapi+(j-1)j/2=aij for 1ij1ij;
  • if uplo = 'L'uplo='L', ap(i + (j1)(2nj) / 2) = aijapi+(j-1)(2n-j)/2=aij for jinjin.
See Section [Further Comments] below for further details.
3:     b(ldb, : :) – complex array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr matrix of right-hand sides BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
The number of linear equations nn, i.e., the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides rr, i.e., the number of columns of the matrix BB.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

ldb

Output Parameters

1:     ap( : :) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
If ifail = 0ifail=0 or n + 1n+1, the factor UU or LL from the Cholesky factorization A = UHUA=UHU or A = LLHA=LLH, in the same storage format as AA.
2:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,n)ldbmax(1,n).
If ifail = 0ifail=0 or n + 1n+1, the nn by rr solution matrix XX.
3:     rcond – double scalar
If ifail = 0ifail=0 or n + 1n+1, an estimate of the reciprocal of the condition number of the matrix AA, computed as rcond = 1 / (A1A11)rcond=1/(A1A-11).
4:     errbnd – double scalar
If ifail = 0ifail=0 or n + 1n+1, an estimate of the forward error bound for a computed solution x^, such that x1 / x1errbndx^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and xx is the corresponding column of the exact solution XX. If rcond is less than machine precision, then errbnd is returned as unity.
5:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail < 0andifail999ifail<0andifail-999
If ifail = iifail=-i, the iith argument had an illegal value.
  ifail = 999ifail=-999
Allocation of memory failed. The double allocatable memory required is n, and the complex allocatable memory required is 2 × n2×n. Allocation failed before the solution could be computed.
  ifail > 0andifailNifail>0andifailN
If ifail = iifail=i, the leading minor of order ii of AA is not positive definite. The factorization could not be completed, and the solution has not been computed.
W ifail = N + 1ifail=N+1
rcond is less than machine precision, so that the matrix AA is numerically singular. A solution to the equations AX = BAX=B has nevertheless been computed.

Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form
(A + E) = b,
(A+E) x^=b,
where
E1 = O(ε) A1
E1=O(ε) A1
and εε is the machine precision. An approximate error bound for the computed solution is given by
(x1)/(x1) κ(A) (E1)/(A1) ,
x^-x1 x1 κ(A) E1 A1 ,
where κ(A) = A11A1κ(A)=A-11A1, the condition number of AA with respect to the solution of the linear equations. nag_linsys_complex_posdef_packed_solve (f04ce) uses the approximation E1 = εA1E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The packed storage scheme is illustrated by the following example when n = 4n=4 and uplo = 'U'uplo='U'. Two-dimensional storage of the Hermitian matrix AA:
a11 a12 a13 a14
a22 a23 a24
a33 a34
a44
(aij = aji)
a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 ( aij = a- ji )
Packed storage of the upper triangle of AA:
ap = a11,a12,a22,a13,a23,a33,a14,a24,a34,a44
[]
ap= [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
The total number of floating point operations required to solve the equations AX = BAX=B is proportional to ((1/3)n3 + n2r)(13n3+n2r). The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of nag_linsys_complex_posdef_packed_solve (f04ce) is nag_linsys_real_posdef_packed_solve (f04be).

Example

function nag_linsys_complex_posdef_packed_solve_example
uplo = 'U';
ap = [3.23;
      1.51 - 1.92i;
      3.58 + 0i;
      1.9 + 0.84i;
      -0.23 + 1.11i;
      4.09 + 0i;
      0.42 + 2.5i;
      -1.18 + 1.37i;
      2.33 - 0.14i;
      4.29 + 0i];
b = [ 3.93 - 6.14i,  1.48 + 6.58i;
      6.17 + 9.42i,  4.65 - 4.75i;
      -7.17 - 21.83i,  -4.91 + 2.29i;
      1.99 - 14.38i,  7.64 - 10.79i];
[apOut, bOut, rcond, errbnd, ifail] = nag_linsys_complex_posdef_packed_solve(uplo, ap, b)
 

apOut =

   1.7972 + 0.0000i
   0.8402 - 1.0683i
   1.3164 + 0.0000i
   1.0572 + 0.4674i
  -0.4702 - 0.3131i
   1.5604 + 0.0000i
   0.2337 + 1.3910i
   0.0834 - 0.0368i
   0.9360 - 0.9900i
   0.6603 + 0.0000i


bOut =

   1.0000 - 1.0000i  -1.0000 + 2.0000i
  -0.0000 + 3.0000i   3.0000 - 4.0000i
  -4.0000 - 5.0000i  -2.0000 + 3.0000i
   2.0000 + 1.0000i   4.0000 - 5.0000i


rcond =

    0.0066


errbnd =

   1.6806e-14


ifail =

                    0


function f04ce_example
uplo = 'U';
ap = [3.23;
      1.51 - 1.92i;
      3.58 + 0i;
      1.9 + 0.84i;
      -0.23 + 1.11i;
      4.09 + 0i;
      0.42 + 2.5i;
      -1.18 + 1.37i;
      2.33 - 0.14i;
      4.29 + 0i];
b = [ 3.93 - 6.14i,  1.48 + 6.58i;
      6.17 + 9.42i,  4.65 - 4.75i;
      -7.17 - 21.83i,  -4.91 + 2.29i;
      1.99 - 14.38i,  7.64 - 10.79i];
[apOut, bOut, rcond, errbnd, ifail] = f04ce(uplo, ap, b)
 

apOut =

   1.7972 + 0.0000i
   0.8402 - 1.0683i
   1.3164 + 0.0000i
   1.0572 + 0.4674i
  -0.4702 - 0.3131i
   1.5604 + 0.0000i
   0.2337 + 1.3910i
   0.0834 - 0.0368i
   0.9360 - 0.9900i
   0.6603 + 0.0000i


bOut =

   1.0000 - 1.0000i  -1.0000 + 2.0000i
  -0.0000 + 3.0000i   3.0000 - 4.0000i
  -4.0000 - 5.0000i  -2.0000 + 3.0000i
   2.0000 + 1.0000i   4.0000 - 5.0000i


rcond =

    0.0066


errbnd =

   1.6806e-14


ifail =

                    0



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