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NAG Toolbox

NAG Toolbox: nag_lapack_zpotrs (f07fs)

Purpose

nag_lapack_zpotrs (f07fs) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
AX = B ,
AX=B ,
where AA has been factorized by nag_lapack_zpotrf (f07fr).

Syntax

[b, info] = f07fs(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpotrs(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zpotrs (f07fs) is used to solve a complex Hermitian positive definite system of linear equations AX = BAX=B, this function must be preceded by a call to nag_lapack_zpotrf (f07fr) which computes the Cholesky factorization of AA. The solution XX is computed by forward and backward substitution.
If uplo = 'U'uplo='U', A = UHUA=UHU, where UU is upper triangular; the solution XX is computed by solving UHY = BUHY=B and then UX = YUX=Y.
If uplo = 'L'uplo='L', A = LLHA=LLH, where LL is lower triangular; the solution XX is computed by solving LY = BLY=B and then LHX = YLHX=Y.

References

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 how AA has been factorized.
uplo = 'U'uplo='U'
A = UHUA=UHU, where UU is upper triangular.
uplo = 'L'uplo='L'
A = LLHA=LLH, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     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 Cholesky factor of AA, as returned by nag_lapack_zpotrf (f07fr).
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 right-hand side matrix BB.

Optional Input Parameters

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

Input Parameters Omitted from the MATLAB Interface

lda ldb

Output Parameters

1:     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).
The nn by rr solution matrix XX.
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: nrhs_p, 4: a, 5: lda, 6: b, 7: ldb, 8: 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.

Accuracy

For each right-hand side vector bb, the computed solution xx is the exact solution of a perturbed system of equations (A + E)x = b(A+E)x=b, where c(n)c(n) is a modest linear function of nn, and εε is the machine precision
If x^ is the true solution, then the computed solution xx satisfies a forward error bound of the form
(x)/(x)c(n)cond(A,x)ε
x-x^ x c(n)cond(A,x)ε
where cond(A,x) = |A1||A||x| / xcond(A) = |A1||A|κ(A)cond(A,x)=|A-1||A||x|/xcond(A)=|A-1||A|κ(A).
Note that cond(A,x)cond(A,x) can be much smaller than cond(A)cond(A).
Forward and backward error bounds can be computed by calling nag_lapack_zporfs (f07fv), and an estimate for κ(A)κ(A) ( = κ1(A)=κ1(A)) can be obtained by calling nag_lapack_zpocon (f07fu).

Further Comments

The total number of real floating point operations is approximately 8n2r8n2r.
This function may be followed by a call to nag_lapack_zporfs (f07fv) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dpotrs (f07fe).

Example

function nag_lapack_zpotrs_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
      1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
      1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 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];
[a, info] = nag_lapack_zpotrf(uplo, a);
[bOut, info] = nag_lapack_zpotrs(uplo, a, b)
 

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


info =

                    0


function f07fs_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
      1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
      1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 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];
[a, info] = f07fr(uplo, a);
[bOut, info] = f07fs(uplo, a, b)
 

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


info =

                    0



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Chapter Introduction
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