NAG Library Routine Document
F07MPF (ZHESVX)
1 Purpose
F07MPF (ZHESVX) uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations
where
A is an
n by
n Hermitian matrix and
X and
B are
n by
r matrices. Error bounds on the solution and a condition estimate are also provided.
2 Specification
SUBROUTINE F07MPF ( |
FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO) |
INTEGER |
N, NRHS, LDA, LDAF, IPIV(*), LDB, LDX, LWORK, INFO |
REAL (KIND=nag_wp) |
RCOND, FERR(*), BERR(*), RWORK(*) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
FACT, UPLO |
|
The routine may be called by its
LAPACK
name zhesvx.
3 Description
F07MPF (ZHESVX) performs the following steps:
- If FACT='N', the diagonal pivoting method is used to factor A. The form of the factorization is A=UDUH if UPLO='U' or A=LDLH if UPLO='L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1 by 1 and 2 by 2 diagonal blocks.
- If some dii=0, so that D is exactly singular, then the routine returns with INFO=i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO=N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
- The system of equations is solved for X using the factored form of A.
- Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002)
Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5 Parameters
- 1: FACT – CHARACTER(1)Input
On entry: specifies whether or not the factorized form of the matrix
A has been supplied.
- FACT='F'
- AF and IPIV contain the factorized form of the matrix A. AF and IPIV will not be modified.
- FACT='N'
- The matrix A will be copied to AF and factorized.
Constraint:
FACT='F' or 'N'.
- 2: UPLO – CHARACTER(1)Input
On entry: if
UPLO='U', the upper triangle of
A is stored.
If UPLO='L', the lower triangle of A is stored.
Constraint:
UPLO='U' or 'L'.
- 3: N – INTEGERInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint:
N≥0.
- 4: NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 5: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the
n by
n Hermitian matrix
A.
- If UPLO='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
- If UPLO='L', the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
- 6: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F07MPF (ZHESVX) is called.
Constraint:
LDA≥max1,N.
- 7: AF(LDAF,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
AF
must be at least
max1,N.
On entry: if
FACT='F',
AF contains the block diagonal matrix
D and the multipliers used to obtain the factor
U or
L from the factorization
A=UDUH or
A=LDLH as computed by
F07MRF (ZHETRF).
On exit: if
FACT='N',
AF returns the block diagonal matrix
D and the multipliers used to obtain the factor
U or
L from the factorization
A=UDUH or
A=LDLH.
- 8: LDAF – INTEGERInput
On entry: the first dimension of the array
AF as declared in the (sub)program from which F07MPF (ZHESVX) is called.
Constraint:
LDAF≥max1,N.
- 9: IPIV(*) – INTEGER arrayInput/Output
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: if
FACT='F',
IPIV contains details of the interchanges and the block structure of
D, as determined by
F07MRF (ZHETRF).
- if IPIVi=k>0, dii is a 1 by 1 pivot block and the ith row and column of A were interchanged with the kth row and column;
- if UPLO='U' and IPIVi-1=IPIVi=-l<0, di-1,i-1d-i,i-1 d-i,i-1dii is a 2 by 2 pivot block and the i-1th row and column of A were interchanged with the lth row and column;
- if UPLO='L' and IPIVi=IPIVi+1=-m<0, diidi+1,idi+1,idi+1,i+1 is a 2 by 2 pivot block and the i+1th row and column of A were interchanged with the mth row and column.
On exit: if
FACT='N',
IPIV contains details of the interchanges and the block structure of
D, as determined by
F07MRF (ZHETRF), as described above.
- 10: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput
-
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.
- 11: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07MPF (ZHESVX) is called.
Constraint:
LDB≥max1,N.
- 12: X(LDX,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
X
must be at least
max1,NRHS.
On exit: if INFO=0 or N+1, the n by r solution matrix X.
- 13: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F07MPF (ZHESVX) is called.
Constraint:
LDX≥max1,N.
- 14: RCOND – REAL (KIND=nag_wp)Output
On exit: the estimate of the reciprocal condition number of the matrix
A. If
RCOND=0.0, the matrix may be exactly singular. This condition is indicated by
INFO>0 and INFO≤N. Otherwise, if
RCOND is less than the
machine precision, the matrix is singular to working precision. This condition is indicated by
INFO=N+1.
- 15: FERR(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
FERR
must be at least
max1,NRHS.
On exit: if
INFO=0 or
N+1, an estimate of the forward error bound for each computed solution vector, such that
x^j-xj∞/xj∞≤FERRj where
x^j is the
jth column of the computed solution returned in the array
X and
xj is the corresponding column of the exact solution
X. The estimate is as reliable as the estimate for
RCOND, and is almost always a slight overestimate of the true error.
- 16: BERR(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
BERR
must be at least
max1,NRHS.
On exit: if INFO=0 or N+1, an estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
- 17: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 returns the optimal
LWORK.
- 18: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F07MPF (ZHESVX) is called.
LWORK≥max1,2×N, and for best performance, when
FACT='N',
LWORK≥max1,2×N,N×nb, where
nb is the optimal block size for
F07MRF (ZHETRF).
If
LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the
WORK array, returns this value as the first entry of the
WORK array, and no error message related to
LWORK is issued.
- 19: RWORK(*) – REAL (KIND=nag_wp) arrayWorkspace
-
Note: the dimension of the array
RWORK
must be at least
max1,N.
- 20: 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 argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0 and INFO≤N
If INFO≤N, di,i is exactly zero. The factorization has been completed, but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND=0.0 is returned.
- INFO=N+1
D is nonsingular, but
RCOND is less than
machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of
RCOND would suggest.
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
where
ε is the
machine precision. See Chapter 11 of
Higham (2002) for further details.
If
x^ is the true solution, then the computed solution
x satisfies a forward error bound of the form
where
condA,x^,b
=
A-1
A
x^
+
b
∞/
x^∞
≤
condA
=
A-1
A
∞≤κ∞
A.
If
x^
is the
j
th column of
X
, then
wc
is returned in
BERRj
and a bound on
x
-
x^
∞
/
x^
∞
is returned in
FERRj
. See Section 4.4 of
Anderson et al. (1999) for further details.
8 Further Comments
The factorization of
A
requires approximately
43
n3
floating point operations.
For each right-hand side, computation of the backward error involves a minimum of
16n2
floating point operations. Each step of iterative refinement involves an additional
24n2
operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form
Ax=b
; the number is usually 4 or 5 and never more than 11. Each solution involves approximately
8n2
operations.
The real analogue of this routine is
F07MBF (DSYSVX). The complex symmetric analogue of this routine is
F07NPF (ZSYSVX).
9 Example
This example solves the equations
where
A
is the Hermitian matrix
and
Error estimates for the solutions, and an estimate of the reciprocal of the condition number of the matrix
A
are also output.
9.1 Program Text
Program Text (f07mpfe.f90)
9.2 Program Data
Program Data (f07mpfe.d)
9.3 Program Results
Program Results (f07mpfe.r)