NAG Library Routine Document
F07ASF (ZGETRS)
1 Purpose
F07ASF (ZGETRS) solves a complex system of linear equations with multiple right-hand sides,
where
A has been factorized by
F07ARF (ZGETRF).
2 Specification
| SUBROUTINE F07ASF ( | TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO) |
| INTEGER | N, NRHS, LDA, IPIV(*), LDB, INFO |
| complex*16 | A(LDA,*), B(LDB,*) |
| CHARACTER*1 | TRANS |
The routine may be called by its
LAPACK
name zgetrs.
3 Description
F07ASF (ZGETRS) is used to solve a complex system of linear equations
AX=B,
ATX=B or
AHX=B, the routine must be preceded by a call to
F07ARF (ZGETRF) which computes the
LU factorization of
A as
A=PLU. The solution is computed by forward and backward substitution.
If TRANS='N', the solution is computed by solving PLY=B and then UX=Y.
If TRANS='T', the solution is computed by solving UTY=B and then LTPTX=Y.
If TRANS='C', the solution is computed by solving UHY=B and then LHPTX=Y.
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: TRANS – CHARACTER*1Input
-
On entry: indicates the form of the equations.
- TRANS='N'
- AX=B is solved for X.
- TRANS='T'
- ATX=B is solved for X.
- TRANS='C'
- AHX=B is solved for X.
Constraint:
TRANS='N', 'T' or 'C'.
- 2: N – INTEGERInput
-
On entry:
n, the order of the matrix A.
Constraint:
N≥0.
- 3: NRHS – INTEGERInput
-
On entry:
r, the number of right-hand sides.
Constraint:
NRHS≥0.
- 4: A(LDA,*) – complex*16 arrayInput
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the
LU factorization of
A, as returned by
F07ARF (ZGETRF).
- 5: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F07ASF (ZGETRS) is called.
Constraint:
LDA≥max1,N.
- 6: IPIV(*) – INTEGER arrayInput
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: the pivot indices, as returned by
F07ARF (ZGETRF).
- 7: B(LDB,*) – complex*16 arrayInput/Output
Note: the second dimension of the array
B
must be at least
max1,NRHS.
On entry: the n by r right-hand side matrix B.
On exit: the n by r solution matrix X.
- 8: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F07ASF (ZGETRS) is called.
Constraint:
LDB≥max1,N.
- 9: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
7 Accuracy
For each right-hand side vector
b, the computed solution
x is the exact solution of a perturbed system of equations
A+Ex=b, where
cn is a modest linear function of
n, and
ε is the
machine precision.
If
x^ is the true solution, then the computed solution
x satisfies a forward error bound of the form
where
condA,x
=
A-1
A
x
∞
/
x∞
≤
condA
=
A-1
A
∞
≤
κ∞
A
.
Note that condA,x can be much smaller than condA, and condAH (which is the same as condAT) can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling
F07AVF (ZGERFS), and an estimate for
κ∞A can be obtained by calling
F07AUF (ZGECON) with
NORM='I'.
8 Further Comments
The total number of real floating-point operations is approximately 8n2r.
This routine may be followed by a call to
F07AVF (ZGERFS) to refine the solution and return an error estimate.
The real analogue of this routine is
F07AEF (DGETRS).
9 Example
This example solves the system of equations
AX=B, where
and
Here
A is nonsymmetric and must first be factorized by
F07ARF (ZGETRF).
9.1 Program Text
Program Text (f07asfe.f)
9.2 Program Data
Program Data (f07asfe.d)
9.3 Program Results
Program Results (f07asfe.r)