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NAG Toolbox: nag_lapack_zsytrs (f07ns)
Purpose
nag_lapack_zsytrs (f07ns) solves a complex symmetric system of linear equations with multiple right-hand sides,
where
has been factorized by
nag_lapack_zsytrf (f07nr).
Syntax
Description
nag_lapack_zsytrs (f07ns) is used to solve a complex symmetric system of linear equations
, this function must be preceded by a call to
nag_lapack_zsytrf (f07nr) which computes the Bunch–Kaufman factorization of
.
If , , where is a permutation matrix, is an upper triangular matrix and is a symmetric block diagonal matrix with by and by blocks; the solution is computed by solving and then .
If , , where is a lower triangular matrix; 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 first dimension of the array
a must be at least
.
The second dimension of the array
a must be at least
.
Details of the factorization of
, as returned by
nag_lapack_zsytrf (f07nr).
- 3:
– int64int32nag_int array
-
The dimension of the array
ipiv
must be at least
Details of the interchanges and the block structure of
, as returned by
nag_lapack_zsytrf (f07nr).
- 4:
– 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 arrays
a,
b and the second dimension of the arrays
a,
ipiv.
, 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_zsyrfs (f07nv), and an estimate for
(
) can be obtained by calling
nag_lapack_zsycon (f07nu).
Further Comments
The total number of real floating-point operations is approximately .
This function may be followed by a call to
nag_lapack_zsyrfs (f07nv) to refine the solution and return an error estimate.
The real analogue of this function is
nag_lapack_dsytrs (f07me).
Example
This example solves the system of equations
, where
and
Here
is symmetric and must first be factorized by
nag_lapack_zsytrf (f07nr).
Open in the MATLAB editor:
f07ns_example
function f07ns_example
fprintf('f07ns example results\n\n');
uplo = 'L';
a = [-0.39 - 0.71i, 0 + 0i, 0 + 0i, 0 + 0i;
5.14 - 0.64i, 8.86 + 1.81i, 0 + 0i, 0 + 0i;
-7.86 - 2.96i, -3.52 + 0.58i, -2.83 - 0.03i, 0 + 0i;
3.80 + 0.92i, 5.32 - 1.59i, -1.54 - 2.86i, -0.56 + 0.12i];
%Factorize A
[af, ipiv, info] = f07nr( ...
uplo, a);
b = [ -55.64 + 41.22i, -19.09 - 35.97i;
-48.18 + 66.00i, -12.08 - 27.02i;
-0.49 - 1.47i, 6.95 + 20.49i;
-6.43 + 19.24i, -4.59 - 35.53i];
[x, info] = f07ns( ...
uplo, af, ipiv, b);
disp('Solution:');
disp(x);
f07ns example results
Solution:
1.0000 - 1.0000i -2.0000 - 1.0000i
-2.0000 + 5.0000i 1.0000 - 3.0000i
3.0000 - 2.0000i 3.0000 + 2.0000i
-4.0000 + 3.0000i -1.0000 + 1.0000i
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