NAG Library Routine Document
F07CBF (DGTSVX)
1 Purpose
F07CBF (DGTSVX) uses the
LU factorization to compute the solution to a real system of linear equations
where
A is a tridiagonal matrix of order
n and
X and
B are
n by
r matrices. Error bounds on the solution and a condition estimate are also provided.
2 Specification
SUBROUTINE F07CBF ( |
FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) |
INTEGER |
N, NRHS, IPIV(*), LDB, LDX, IWORK(N), INFO |
REAL (KIND=nag_wp) |
DL(*), D(*), DU(*), DLF(*), DF(*), DUF(*), DU2(*), B(LDB,*), X(LDX,*), RCOND, FERR(NRHS), BERR(NRHS), WORK(3*N) |
CHARACTER(1) |
FACT, TRANS |
|
The routine may be called by its
LAPACK
name dgtsvx.
3 Description
F07CBF (DGTSVX) performs the following steps:
- If FACT='N', the LU decomposition is used to factor the matrix A as A=LU, where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
- If some uii=0, so that U 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'
- DLF, DF, DUF, DU2 and IPIV contain the factorized form of the matrix A. DLF, DF, DUF, DU2 and IPIV will not be modified.
- FACT='N'
- The matrix A will be copied to DLF, DF and DUF and factorized.
Constraint:
FACT='F' or 'N'.
- 2: TRANS – CHARACTER(1)Input
On entry: specifies the form of the system of equations.
- TRANS='N'
- AX=B (No transpose).
- TRANS='T' or 'C'
- ATX=B (Transpose).
Constraint:
TRANS='N', 'T' or 'C'.
- 3: N – INTEGERInput
On entry: n, 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: DL(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DL
must be at least
max1,N-1.
On entry: the n-1 subdiagonal elements of A.
- 6: D(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: the n diagonal elements of A.
- 7: DU(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DU
must be at least
max1,N-1.
On entry: the n-1 superdiagonal elements of A.
- 8: DLF(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
DLF
must be at least
max1,N-1.
On entry: if
FACT='F',
DLF contains the
n-1 multipliers that define the matrix
L from the
LU factorization of
A.
On exit: if
FACT='N',
DLF contains the
n-1 multipliers that define the matrix
L from the
LU factorization of
A.
- 9: DF(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
DF
must be at least
max1,N.
On entry: if
FACT='F',
DF contains the
n diagonal elements of the upper triangular matrix
U from the
LU factorization of
A.
On exit: if
FACT='N',
DF contains the
n diagonal elements of the upper triangular matrix
U from the
LU factorization of
A.
- 10: DUF(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
DUF
must be at least
max1,N-1.
On entry: if
FACT='F',
DUF contains the
n-1 elements of the first superdiagonal of
U.
On exit: if
FACT='N',
DUF contains the
n-1 elements of the first superdiagonal of
U.
- 11: DU2(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
DU2
must be at least
max1,N-2.
On entry: if
FACT='F',
DU2 contains the (
n-2) elements of the second superdiagonal of
U.
On exit: if
FACT='N',
DU2 contains the (
n-2) elements of the second superdiagonal of
U.
- 12: IPIV(*) – INTEGER arrayInput/Output
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: if
FACT='F',
IPIV contains the pivot indices from the
LU factorization of
A.
On exit: if
FACT='N',
IPIV contains the pivot indices from the
LU factorization of
A; row
i of the matrix was interchanged with row
IPIVi.
IPIVi will always be either
i or
i+1;
IPIVi=i indicates a row interchange was not required.
- 13: B(LDB,*) – REAL (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.
- 14: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07CBF (DGTSVX) is called.
Constraint:
LDB≥max1,N.
- 15: X(LDX,*) – REAL (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.
- 16: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F07CBF (DGTSVX) is called.
Constraint:
LDX≥max1,N.
- 17: 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.
- 18: FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
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.
- 19: BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
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).
- 20: WORK(3×N) – REAL (KIND=nag_wp) arrayWorkspace
- 21: IWORK(N) – INTEGER arrayWorkspace
- 22: 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=i, ui,i is exactly zero. The factorization has not been completed unless i=N, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND=0.0 is returned.
- INFO=N+1
The triangular matrix
U 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
cn is a modest linear function of
n, and
ε is the
machine precision. See Section 9.3 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 total number of floating point operations required to solve the equations
AX=B
is proportional to
nr
.
The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The complex analogue of this routine is
F07CPF (ZGTSVX).
9 Example
This example solves the equations
where
A
is the tridiagonal matrix
and
Estimates for the backward errors, forward errors and condition number are also output.
9.1 Program Text
Program Text (f07cbfe.f90)
9.2 Program Data
Program Data (f07cbfe.d)
9.3 Program Results
Program Results (f07cbfe.r)