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NAG Toolbox: nag_lapack_dpftrs (f07we)
Purpose
nag_lapack_dpftrs (f07we) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
using the Cholesky factorization computed by
nag_lapack_dpftrf (f07wd) stored in Rectangular Full Packed (RFP) format.
Syntax
Description
nag_lapack_dpftrs (f07we) is used to solve a real symmetric positive definite system of linear equations
, the function must be preceded by a call to
nag_lapack_dpftrf (f07wd) which computes the Cholesky factorization of
, stored in RFP format.
The RFP storage format is described in
Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.
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
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2
Parameters
Compulsory Input Parameters
- 1:
– string (length ≥ 1)
-
Specifies whether the RFP representation of
is normal or transposed.
- The matrix is stored in normal RFP format.
- The matrix is stored in transposed RFP format.
Constraint:
or .
- 2:
– string (length ≥ 1)
-
Specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
- 3:
– double array
-
The Cholesky factorization of
stored in RFP format, as returned by
nag_lapack_dpftrf (f07wd).
- 4:
– double 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
ar.
, 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:
– double 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
and
is the condition number when using the
-norm.
Note that can be much smaller than .
Further Comments
The total number of floating-point operations is approximately .
The complex analogue of this function is
nag_lapack_zpftrs (f07ws).
Example
This example solves the system of equations
, where
Here
is symmetric positive definite, stored in RFP format, and must first be factorized by
nag_lapack_dpftrf (f07wd).
Open in the MATLAB editor:
f07we_example
function f07we_example
fprintf('f07we example results\n\n');
transr = 'n';
uplo = 'l';
ar = [ 0.76 0.34;
4.16 1.18;
-3.12 5.03;
0.56 -0.83;
-0.10 1.18];
n = int64(4);
n2 = (n*(n+1))/2;
ar = reshape(ar,[n2,1]);
b = [ 8.70, 8.30;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5.00];
[ar, info] = f07wd(transr, uplo, n, ar);
if info == 0
[b, info] = f07we(transr, uplo, ar, b);
fprintf('\n');
[ifail] = x04ca('g', ' ', b, 'Solutions');
else
fprintf('\na is not positive definite.\n');
end
f07we example results
Solutions
1 2
1 1.0000 4.0000
2 -1.0000 3.0000
3 2.0000 2.0000
4 -3.0000 1.0000
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