*DECK SNBFA SUBROUTINE SNBFA (ABE, LDA, N, ML, MU, IPVT, INFO) C***BEGIN PROLOGUE SNBFA C***PURPOSE Factor a real band matrix by elimination. C***LIBRARY SLATEC C***CATEGORY D2A2 C***TYPE SINGLE PRECISION (SNBFA-S, DNBFA-D, CNBFA-C) C***KEYWORDS BANDED, LINEAR EQUATIONS, MATRIX FACTORIZATION, C NONSYMMETRIC C***AUTHOR Voorhees, E. A., (LANL) C***DESCRIPTION C C SNBFA factors a real band matrix by elimination. C C SNBFA is usually called by SNBCO, but it can be called C directly with a saving in time if RCOND is not needed. C C On Entry C C ABE REAL(LDA, NC) C contains the matrix in band storage. The rows C of the original matrix are stored in the rows C of ABE and the diagonals of the original matrix C are stored in columns 1 through ML+MU+1 of ABE. C NC must be .GE. 2*ML+MU+1 . C See the comments below for details. C C LDA INTEGER C the leading dimension of the array ABE. C LDA must be .GE. N . C C N INTEGER C the order of the original matrix. C C ML INTEGER C number of diagonals below the main diagonal. C 0 .LE. ML .LT. N . C C MU INTEGER C number of diagonals above the main diagonal. C 0 .LE. MU .LT. N . C More efficient if ML .LE. MU . C C On Return C C ABE an upper triangular matrix in band storage C and the multipliers which were used to obtain it. C The factorization can be written A = L*U , where C L is a product of permutation and unit lower C triangular matrices and U is upper triangular. C C IPVT INTEGER(N) C an integer vector of pivot indices. C C INFO INTEGER C =0 normal value C =K if U(K,K) .EQ. 0.0 . This is not an error C condition for this subroutine, but it does C indicate that SNBSL will divide by zero if C called. Use RCOND in SNBCO for a reliable C indication of singularity. C C Band Storage C C If A is a band matrix, the following program segment C will set up the input. C C ML = (band width below the diagonal) C MU = (band width above the diagonal) C DO 20 I = 1, N C J1 = MAX(1, I-ML) C J2 = MIN(N, I+MU) C DO 10 J = J1, J2 C K = J - I + ML + 1 C ABE(I,K) = A(I,J) C 10 CONTINUE C 20 CONTINUE C C This uses columns 1 through ML+MU+1 of ABE . C Furthermore, ML additional columns are needed in C ABE starting with column ML+MU+2 for elements C generated during the triangularization. The total C number of columns needed in ABE is 2*ML+MU+1 . C C Example: If the original matrix is C C 11 12 13 0 0 0 C 21 22 23 24 0 0 C 0 32 33 34 35 0 C 0 0 43 44 45 46 C 0 0 0 54 55 56 C 0 0 0 0 65 66 C C then N = 6, ML = 1, MU = 2, LDA .GE. 5 and ABE should contain C C * 11 12 13 + , * = not used C 21 22 23 24 + , + = used for pivoting C 32 33 34 35 + C 43 44 45 46 + C 54 55 56 * + C 65 66 * * + C C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. C Stewart, LINPACK Users' Guide, SIAM, 1979. C***ROUTINES CALLED ISAMAX, SAXPY, SSCAL, SSWAP C***REVISION HISTORY (YYMMDD) C 800606 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE SNBFA INTEGER LDA,N,ML,MU,IPVT(*),INFO REAL ABE(LDA,*) C INTEGER ML1,MB,M,N1,LDB,I,J,K,L,LM,LM1,LM2,MP,ISAMAX REAL T C***FIRST EXECUTABLE STATEMENT SNBFA ML1=ML+1 MB=ML+MU M=ML+MU+1 N1=N-1 LDB=LDA-1 INFO=0 C C SET FILL-IN COLUMNS TO ZERO C IF(N.LE.1)GO TO 50 IF(ML.LE.0)GO TO 7 DO 6 J=1,ML DO 5 I=1,N ABE(I,M+J)=0.0E0 5 CONTINUE 6 CONTINUE 7 CONTINUE C C GAUSSIAN ELIMINATION WITH PARTIAL ELIMINATION C DO 40 K=1,N1 LM=MIN(N-K,ML) LM1=LM+1 LM2=ML1-LM C C SEARCH FOR PIVOT INDEX C L=-ISAMAX(LM1,ABE(LM+K,LM2),LDB)+LM1+K IPVT(K)=L MP=MIN(MB,N-K) C C SWAP ROWS IF NECESSARY C IF(L.NE.K)CALL SSWAP(MP+1,ABE(K,ML1),LDA,ABE(L,ML1+K-L),LDA) C C SKIP COLUMN REDUCTION IF PIVOT IS ZERO C IF(ABE(K,ML1).EQ.0.0E0) GO TO 20 C C COMPUTE MULTIPLIERS C T=-1.0/ABE(K,ML1) CALL SSCAL(LM,T,ABE(LM+K,LM2),LDB) C C ROW ELIMINATION WITH COLUMN INDEXING C DO 10 J=1,MP CALL SAXPY (LM,ABE(K,ML1+J),ABE(LM+K,LM2),LDB,ABE(LM+K,LM2+J), 1 LDB) 10 CONTINUE GO TO 30 20 CONTINUE INFO=K 30 CONTINUE 40 CONTINUE 50 CONTINUE IPVT(N)=N IF(ABE(N,ML1).EQ.0.0E0) INFO=N RETURN END