NAG FL Interface
f01lhf (real_​gen_​blkdiag_​lu)

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1 Purpose

f01lhf factorizes a real almost block diagonal matrix.

2 Specification

Fortran Interface
Subroutine f01lhf ( n, nbloks, blkstr, a, lena, pivot, tol, kpivot, ifail)
Integer, Intent (In) :: n, nbloks, blkstr(3,nbloks), lena
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: pivot(n), kpivot
Real (Kind=nag_wp), Intent (Inout) :: a(lena), tol
C Header Interface
#include <nag.h>
void  f01lhf_ (const Integer *n, const Integer *nbloks, const Integer blkstr[], double a[], const Integer *lena, Integer pivot[], double *tol, Integer *kpivot, Integer *ifail)
The routine may be called by the names f01lhf or nagf_matop_real_gen_blkdiag_lu.

3 Description

f01lhf factorizes a real almost block diagonal matrix, A, by row elimination with alternate row and column pivoting such that no ‘fill-in’ is produced. The code, which is derived from ARCECO described in Diaz et al. (1983), uses Level 1 and Level 2 BLAS. No three successive diagonal blocks may have columns in common and, therefore, the almost block diagonal matrix must have the form shown in the following diagram:
Figure 1
Figure 1
This routine may be followed by f04lhf, which is designed to solve sets of linear equations AX=B or ATX=B.

4 References

Diaz J C, Fairweather G and Keast P (1983) Fortran packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination ACM Trans. Math. Software 9 358–375

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n>0.
2: nbloks Integer Input
On entry: n, the total number of blocks of the matrix A.
Constraint: 0<nbloksn.
3: blkstr(3,nbloks) Integer array Input
On entry: information which describes the block structure of A as follows:
  • blkstr(1,k) must contain the number of rows in the kth block, k=1,2,,nbloks;
  • blkstr(2,k) must contain the number of columns in the kth block, k=1,2,,nbloks;
  • blkstr(3,k) must contain the number of columns of overlap between the kth and (k+1)th blocks, k=1,2,,nbloks-1. blkstr(3,nbloks) need not be set.
The following conditions delimit the structure of A:
  • blkstr(1,k),blkstr(2,k)>0,  k=1,2,,nbloks,
  • blkstr(3,k)0,   k=1,2,,nbloks-1,
(there must be at least one column and one row in each block and a non-negative number of columns of overlap);
  • blkstr(3,k-1)+blkstr(3,k)blkstr(2,k),  k=2,3,,nbloks-1,
(the total number of columns in overlaps in each block must not exceed the number of columns in that block);
  • blkstr(2,1)blkstr(1,1),
  • blkstr(2,1)+ k =2 j [blkstr(2,k)-blkstr(3,k -1 )] k =1 j blkstr(1,k), j=2,3,,nbloks-1 ,
  • k=1j[blkstr(2,k)-blkstr(3,k)]k=1jblkstr(1,k),  j=1,2,,nbloks-1,
(the index of the first column of the overlap between the jth and (j+1)th blocks must be the index of the last row of the jth block, and the index of the last column of overlap must be the index of the last row of the jth block);
  • k =1 nbloks blkstr(1,k)=n ,
  • blkstr(2,1)+ k =2 nbloks [blkstr(2,k)-blkstr(3,k -1 )] =nk ,
(both the number of rows and the number of columns of A must equal n).
4: a(lena) Real (Kind=nag_wp) array Input/Output
On entry: the elements of the almost block diagonal matrix stored block by block, with each block stored column by column. The sizes of the blocks and the overlaps are defined by the argument blkstr.
If ars is the first element in the kth block, then an arbitrary element aij in the kth block must be stored in the array element:
a( pk+ (j-r) mk+ (i-s)+1 )  
pk=l= 1 k- 1blkstr(1,l)×blkstr(2,l)  
is the base address of the kth block, and
is the number of rows of the kth block.
See Section 9 for comments on scaling.
On exit: the factorized form of the matrix.
5: lena Integer Input
On entry: the dimension of the array a as declared in the (sub)program from which f01lhf is called.
Constraint: lena k =1 nbloks [blkstr(1,k)×blkstr(2,k)] .
6: pivot(n) Integer array Output
On exit: details of the interchanges.
7: tol Real (Kind=nag_wp) Input/Output
On entry: a relative tolerance to be used to indicate whether or not the matrix is singular. For a discussion on how tol is used see Section 9. If tol is non-positive, tol is reset to 10ε, where ε is the machine precision.
On exit: unchanged unless tol0.0 on entry, in which case it is set to 10ε.
8: kpivot Integer Output
On exit: if ifail=2, kpivot contains the value k, where k is the first position on the diagonal of the matrix A where too small a pivot was detected. Otherwise kpivot is set to 0.
9: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, K=value, blkstr(2,K)=value and blkstr(1,K)=value.
Constraint: blkstr(2,K)blkstr(1,K).
On entry, K=value, blkstr(2,K)=value blkstr(3,K)=value and blkstr(1,K)=value.
Constraint: blkstr(2,K)-blkstr(3,K)blkstr(1,K).
On entry, K=value and blkstr(1,K)=value.
Constraint: blkstr(1,K)1.
On entry, K=value and blkstr(2,K)=value.
Constraint: blkstr(2,K)1.
On entry, K=value and blkstr(3,K)=value.
Constraint: blkstr(3,K)0.
On entry, K=value blkstr(3,K)=value blkstr(3,K-1)=value and blkstr(2,K)=value.
Constraint: blkstr(3,K)+blkstr(3,K-1)blkstr(2,K).
On entry, lena is too small. lena=value. Minimum possible dimension: value.
On entry, n=value.
Constraint: n1.
On entry, n=value and nbloks=value.
Constraint: nnbloks.
On entry, nbloks=value.
Constraint: nbloks1.
On entry, the following equality does not hold: blkstr(2,1)+sum(blkstr(2,k)-blkstr(3,k-1):k=2,nbloks)=n.
On entry, the following equality does not hold: sum(blkstr(1,k):k=1,nbloks)=n.
On entry, the following inequality was not satisfied for: J=value. sum(blkstr(2,k)-blkstr(3,k):k=1,J) sum(blkstr(1,k):k=1,J) blkstr(2,1)+sum(blkstr(2,k)-blkstr(3,k-1):k=2,J).
Factorization completed, but pivot in diagonal I was small: I=value.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy of f01lhf depends on the conditioning of the matrix A.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01lhf 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.

9 Further Comments

Singularity or near singularity in A is determined by the argument tol. If the absolute value of any pivot is less than tol×amax, where amax is the maximum absolute value of an element of A, then A is said to be singular. The position on the diagonal of A of the first of any such pivots is indicated by the argument kpivot. The factorization, and the test for near singularity, will be more accurate if before entry A is scaled so that the -norms of the rows and columns of A are all of approximately the same order of magnitude. (The -norm is the maximum absolute value of any element in the row or column.)

10 Example

This example solves the set of linear equations Ax=b where
A= ( -1.00 -0.98 -0.79 -0.15 -1.00 -0.25 -0.87 0.35 0.78 0.31 -0.85 0.89 -0.69 -0.98 -0.76 -0.82 0.12 -0.01 0.75 0.32 -1.00 -0.53 -0.83 -0.98 -0.58 0.04 0.87 0.38 -1.00 -0.21 -0.93 -0.84 0.37 -0.94 -0.96 -1.00 -0.99 -0.91 -0.28 -0.90 0.78 -0.93 -0.76 0.48 -0.87 -0.14 -1.00 -0.59 -0.99 0.21 -0.73 -0.48 -0.93 -0.91 0.10 -0.89 -0.68 -0.09 -0.58 -0.21 0.85 -0.39 0.79 -0.71 0.39 -0.99 -0.12 -0.75 0.17 -1.37 1.29 -1.59 1.10 -1.63 -1.01 -0.27 0.08 0.61 0.54 -0.41 0.16 -0.46 -0.67 0.56 -0.99 0.16 -0.16 0.98 -0.24 -0.41 0.40 -0.93 0.70 0.43 0.71 -0.97 -0.60 -0.30 0.18 -0.47 -0.98 -0.73 0.07 0.04 -0.25 -0.92 -0.52 -0.46 -0.58 -0.89 -0.94 -0.54 -1.00 -0.36 )  
b= ( -2.92 -1.17 -1.30 -1.17 -2.10 -4.51 -1.71 -4.59 -4.19 -0.93 -3.31 0.52 -0.12 -0.05 -0.98 -2.07 -2.73 -1.95 )  
The exact solution is

10.1 Program Text

Program Text (f01lhfe.f90)

10.2 Program Data

Program Data (f01lhfe.d)

10.3 Program Results

Program Results (f01lhfe.r)