The function may be called by the names: f07crc, nag_lapacklin_zgttrf or nag_zgttrf.
f07crc uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix as
where 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 https://www.netlib.org/lapack/lug
1: – IntegerInput
On entry: , the order of the matrix .
2: – ComplexInput/Output
Note: the dimension, dim, 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: – ComplexInput/Output
Note: the dimension, dim, 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: – ComplexInput/Output
Note: the dimension, dim, 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: – ComplexOutput
On exit: contains the elements of the second superdiagonal of .
6: – IntegerOutput
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: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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
and is the machine precision.
Following the use of this function, f07csc can be used to solve systems of equations or or , and f07cuc can be used to estimate the condition number of .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07crc 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 function. 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 .