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NAG Toolbox: nag_lapack_zpptrs (f07gs)
Purpose
nag_lapack_zpptrs (f07gs) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
where
has been factorized by
nag_lapack_zpptrf (f07gr), using packed storage.
Syntax
Description
nag_lapack_zpptrs (f07gs) is used to solve a complex Hermitian positive definite system of linear equations
, the function must be preceded by a call to
nag_lapack_zpptrf (f07gr) which computes the Cholesky factorization of
, using packed storage. The solution
is computed by forward and backward substitution.
If , , where is upper triangular; the solution is computed by solving and then .
If , , where is lower triangular; the solution is computed by solving and then .
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parameters
Compulsory Input Parameters
- 1:
– string (length ≥ 1)
-
Specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
- 2:
– complex array
-
The dimension of the array
ap
must be at least
The Cholesky factor of
stored in packed form, as returned by
nag_lapack_zpptrf (f07gr).
- 3:
– complex array
-
The first dimension of the array
b must be at least
.
The second dimension of the array
b must be at least
.
The by right-hand side matrix .
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the array
ap and the second dimension of the array
ap. (An error is raised if these dimensions are not equal.)
, the order of the matrix .
Constraint:
.
- 2:
– int64int32nag_int scalar
-
Default:
the second dimension of the array
b.
, the number of right-hand sides.
Constraint:
.
Output Parameters
- 1:
– complex array
-
The first dimension of the array
b will be
.
The second dimension of the array
b will be
.
The by solution matrix .
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
-
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
Accuracy
For each right-hand side vector
, the computed solution
is the exact solution of a perturbed system of equations
, where
- if , ;
- if , ,
is a modest linear function of
, and
is the
machine precisionIf
is the true solution, then the computed solution
satisfies a forward error bound of the form
where
.
Note that can be much smaller than .
Forward and backward error bounds can be computed by calling
nag_lapack_zpprfs (f07gv), and an estimate for
(
) can be obtained by calling
nag_lapack_zppcon (f07gu).
Further Comments
The total number of real floating-point operations is approximately .
This function may be followed by a call to
nag_lapack_zpprfs (f07gv) to refine the solution and return an error estimate.
The real analogue of this function is
nag_lapack_dpptrs (f07ge).
Example
This example solves the system of equations
, where
and
Here
is Hermitian positive definite, stored in packed form, and must first be factorized by
nag_lapack_zpptrf (f07gr).
Open in the MATLAB editor:
f07gs_example
function f07gs_example
fprintf('f07gs example results\n\n');
uplo = 'L';
n = int64(4);
ap = [3.23 + 0i 1.51 + 1.92i 1.90 - 0.84i 0.42 - 2.50i ...
3.58 + 0i -0.23 - 1.11i -1.18 - 1.37i ...
4.09 + 0.00i 2.33 + 0.14i ...
4.29 + 0.00i];
[L, info] = f07gr( ...
uplo, n, ap);
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];
[x, info] = f07gs( ...
uplo, L, b);
disp('Solution(s)');
disp(x);
f07gs example results
Solution(s)
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
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