F07CEF (DGTTRS) computes the solution to a real system of linear equations
AX=B
or
ATX=B
, where
A
is an
n
by
n
tridiagonal matrix and
X
and
B
are
n
by
r
matrices, using the
LU
factorization returned by
F07CDF (DGTTRF).
F07CEF (DGTTRS) should be preceded by a call to
F07CDF (DGTTRF), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix
A
as
where
P
is a permutation matrix,
L
is unit lower triangular with at most one nonzero subdiagonal element in each column, and
U
is an upper triangular band matrix, with two superdiagonals. F07CEF (DGTTRS) then utilizes the factorization to solve the required equations.
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- 1: TRANS – CHARACTER(1)Input
On entry: specifies the equations to be solved as follows:
- TRANS='N'
- Solve AX=B for X.
- TRANS='T' or 'C'
- Solve ATX=B 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, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 4: DL(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DL
must be at least
max1,N-1.
On entry: must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
- 5: D(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
- 6: DU(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DU
must be at least
max1,N-1.
On entry: must contain the n-1 elements of the first superdiagonal of U.
- 7: DU2(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DU2
must be at least
max1,N-2.
On entry: must contain the n-2 elements of the second superdiagonal of U.
- 8: IPIV(*) – INTEGER arrayInput
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row IPIVi, and IPIVi must always be either i or i+1, IPIVi=i indicating that a row interchange was not performed.
- 9: 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 matrix of right-hand sides B.
On exit: the n by r solution matrix X.
- 10: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07CEF (DGTTRS) is called.
Constraint:
LDB≥max1,N.
- 11: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
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.
Following the use of this routine
F07CGF (DGTCON) can be used to estimate the condition number of
A
and
F07CHF (DGTRFS) can be used to obtain approximate error bounds.
The complex analogue of this routine is
F07CSF (ZGTTRS).
This example solves the equations
where
A
is the tridiagonal matrix