NAG Library Routine Document
F07MAF (DSYSV)
1 Purpose
F07MAF (DSYSV) computes the solution to a real system of linear equations
where
A is an
n by
n symmetric matrix and
X and
B are
n by
r matrices.
2 Specification
SUBROUTINE F07MAF ( |
UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO) |
INTEGER |
N, NRHS, LDA, IPIV(*), LDB, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), B(LDB,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dsysv.
3 Description
F07MAF (DSYSV) uses the diagonal pivoting method to factor A as
A=UDUT if UPLO='U' or A=LDLT if UPLO='L',
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. The factored form of A is then used to solve the system of equations AX=B.
Note that, in general, different permutations (pivot sequences) and diagonal block structures are obtained for UPLO='U' or 'L'
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
5 Parameters
- 1: 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'.
- 2: N – INTEGERInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint:
N≥0.
- 3: NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 4: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the
n by
n symmetric 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.
On exit: if
INFO=0, the block diagonal matrix
D and the multipliers used to obtain the factor
U or
L from the factorization
A=UDUT or
A=LDLT as computed by
F07MDF (DSYTRF).
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F07MAF (DSYSV) is called.
Constraint:
LDA≥max1,N.
- 6: IPIV(*) – INTEGER arrayOutput
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On exit: details of the interchanges and the block structure of
D. More precisely,
- 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.
- 7: B(LDB,*) – REAL (KIND=nag_wp) 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: if INFO=0, 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 F07MAF (DSYSV) is called.
Constraint:
LDB≥max1,N.
- 9: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 returns the optimal
LWORK.
- 10: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F07MAF (DSYSV) is called.
LWORK≥1, and for best performance
LWORK≥max1,N×nb, where
nb is the optimal block size for
F07MDF (DSYTRF).
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.
- 11: 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
If INFO=i, dii is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.
7 Accuracy
The computed solution for a single right-hand side,
x^
, satisfies an equation of the form
where
and
ε
is the
machine precision. An approximate error bound for the computed solution is given by
where
κA
=
A-11
A1
, the condition number of
A
with respect to the solution of the linear equations. See Section 4.4 of
Anderson et al. (1999) for further details.
F07MBF (DSYSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively,
F04BHF solves
Ax=b
and returns a forward error bound and condition estimate.
F04BHF calls F07MAF (DSYSV) to solve the equations.
8 Further Comments
The total number of floating point operations is approximately
13
n3
+
2n2r
, where
r
is the number of right-hand sides.
The complex analogues of F07MAF (DSYSV) are
F07MNF (ZHESV) for Hermitian matrices, and
F07NNF (ZSYSV) for symmetric matrices.
9 Example
This example solves the equations
where
A
is the symmetric matrix
Details of the factorization of
A
are also output.
9.1 Program Text
Program Text (f07mafe.f90)
9.2 Program Data
Program Data (f07mafe.d)
9.3 Program Results
Program Results (f07mafe.r)