NAG Library Routine Document
f07crf (zgttrf) computes the factorization of a complex by tridiagonal matrix .
|Integer, Intent (In)||:: ||
|Integer, Intent (Out)||:: ||
|Complex (Kind=nag_wp), Intent (Inout)||:: ||
|Complex (Kind=nag_wp), Intent (Out)||:: ||
du2(n-2)|C Header Interface
const Integer *n,
The routine may be called by its
uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix
is a permutation matrix,
is unit lower triangular with at most one nonzero subdiagonal element in each column, and
is an upper triangular band matrix, with two superdiagonals.
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: – Complex (Kind=nag_wp) arrayInput/Output
the dimension of the array dl
must be at least
On entry: must contain the subdiagonal elements of the matrix .
On exit: is overwritten by the multipliers that define the matrix of the factorization of .
- 3: – Complex (Kind=nag_wp) arrayInput/Output
the dimension of the array d
must be at least
On entry: must contain the diagonal elements of the matrix .
On exit: is overwritten by the diagonal elements of the upper triangular matrix from the factorization of .
- 4: – Complex (Kind=nag_wp) arrayInput/Output
the dimension of the array du
must be at least
On entry: must contain the superdiagonal elements of the matrix .
On exit: is overwritten by the elements of the first superdiagonal of .
- 5: – Complex (Kind=nag_wp) arrayOutput
On exit: contains the elements of the second superdiagonal of .
- 6: – Integer arrayOutput
On exit: contains the pivot indices that define the permutation matrix . At the th step, row of the matrix was interchanged with row . will always be either or , indicating that a row interchange was not performed.
- 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.
Element of the diagonal is exactly zero.
The factorization has been completed, but the factor is exactly singular, and division by zero will occur if it is used to solve
a system of equations.
The computed factorization satisfies an equation of the form
is the machine precision
Following the use of this routine, f07csf (zgttrs)
can be used to solve systems of equations
, and f07cuf (zgtcon)
can be used to estimate the condition number of
Parallelism and Performance
f07crf (zgttrf) 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 factorize the matrix is proportional to .
The real analogue of this routine is f07cdf (dgttrf)
This example factorizes the tridiagonal matrix
Program Text (f07crfe.f90)
Program Data (f07crfe.d)
Program Results (f07crfe.r)