NAG Library Routine Document
computes the solution to a real system of linear equations
symmetric positive definite tridiagonal matrix and
matrices, using the
factorization returned by f07jdf (dpttrf)
|Integer, Intent (In)||:: ||n, nrhs, ldb|
|Integer, Intent (Out)||:: ||info|
|Real (Kind=nag_wp), Intent (In)||:: ||d(*), e(*)|
|Real (Kind=nag_wp), Intent (Inout)||:: ||b(ldb,*)|C Header Interface
f07jef_ (const Integer *n, const Integer *nrhs, const double d, const double e, double b, const Integer *ldb, Integer *info)|
The routine may be called by its
should be preceded by a call to f07jdf (dpttrf)
, which computes a modified Cholesky factorization of the matrix
is a unit lower bidiagonal matrix and
is a diagonal matrix, with positive diagonal elements. f07jef (dpttrs)
then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form
is a unit upper bidiagonal matrix.
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: – IntegerInput
On entry: , the order of the matrix .
- 2: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
- 3: – Real (Kind=nag_wp) arrayInput
the dimension of the array d
must be at least
On entry: must contain the diagonal elements of the diagonal matrix from the factorization of .
- 4: – Real (Kind=nag_wp) arrayInput
the dimension of the array e
must be at least
: must contain the
subdiagonal elements of the unit lower bidiagonal matrix
can also be regarded as the superdiagonal of the unit upper bidiagonal matrix
- 5: – Real (Kind=nag_wp) arrayInput/Output
the second dimension of the array b
must be at least
On entry: the by matrix of right-hand sides .
On exit: the by solution matrix .
- 6: – IntegerInput
: the first dimension of the array b
as declared in the (sub)program from which f07jef (dpttrs)
- 7: – IntegerOutput
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed solution for a single right-hand side,
, satisfies an equation of the form
is the machine precision
. An approximate error bound for the computed solution is given by
, the condition number of
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 f07jgf (dptcon)
can be used to estimate the condition number of
and f07jhf (dptrfs)
can be used to obtain approximate error bounds.
Parallelism and Performance
f07jef (dpttrs) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction
for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note
for your implementation for any additional implementation-specific information.
The total number of floating-point operations required to solve the equations is proportional to .
The complex analogue of this routine is f07jsf (zpttrs)
This example solves the equations
is the symmetric positive definite tridiagonal matrix
Program Text (f07jefe.f90)
Program Data (f07jefe.d)
Program Results (f07jefe.r)