*DECK SGBFA
SUBROUTINE SGBFA (ABD, LDA, N, ML, MU, IPVT, INFO)
C***BEGIN PROLOGUE SGBFA
C***PURPOSE Factor a band matrix using Gaussian elimination.
C***LIBRARY SLATEC (LINPACK)
C***CATEGORY D2A2
C***TYPE SINGLE PRECISION (SGBFA-S, DGBFA-D, CGBFA-C)
C***KEYWORDS BANDED, LINEAR ALGEBRA, LINPACK, MATRIX FACTORIZATION
C***AUTHOR Moler, C. B., (U. of New Mexico)
C***DESCRIPTION
C
C SGBFA factors a real band matrix by elimination.
C
C SGBFA is usually called by SBGCO, but it can be called
C directly with a saving in time if RCOND is not needed.
C
C On Entry
C
C ABD REAL(LDA, N)
C contains the matrix in band storage. The columns
C of the matrix are stored in the columns of ABD and
C the diagonals of the matrix are stored in rows
C ML+1 through 2*ML+MU+1 of ABD .
C See the comments below for details.
C
C LDA INTEGER
C the leading dimension of the array ABD .
C LDA must be .GE. 2*ML + MU + 1 .
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 On Return
C
C ABD an upper triangular matrix in band storage and
C 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 SGBSL will divide by zero if
C called. Use RCOND in SBGCO 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 M = ML + MU + 1
C DO 20 J = 1, N
C I1 = MAX(1, J-MU)
C I2 = MIN(N, J+ML)
C DO 10 I = I1, I2
C K = I - J + M
C ABD(K,J) = A(I,J)
C 10 CONTINUE
C 20 CONTINUE
C
C This uses rows ML+1 through 2*ML+MU+1 of ABD .
C In addition, the first ML rows in ABD are used for
C elements generated during the triangularization.
C The total number of rows needed in ABD is 2*ML+MU+1 .
C The ML+MU by ML+MU upper left triangle and the
C ML by ML lower right triangle are not referenced.
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
C***REVISION HISTORY (YYMMDD)
C 780814 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 900326 Removed duplicate information from DESCRIPTION section.
C (WRB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE SGBFA
INTEGER LDA,N,ML,MU,IPVT(*),INFO
REAL ABD(LDA,*)
C
REAL T
INTEGER I,ISAMAX,I0,J,JU,JZ,J0,J1,K,KP1,L,LM,M,MM,NM1
C
C***FIRST EXECUTABLE STATEMENT SGBFA
M = ML + MU + 1
INFO = 0
C
C ZERO INITIAL FILL-IN COLUMNS
C
J0 = MU + 2
J1 = MIN(N,M) - 1
IF (J1 .LT. J0) GO TO 30
DO 20 JZ = J0, J1
I0 = M + 1 - JZ
DO 10 I = I0, ML
ABD(I,JZ) = 0.0E0
10 CONTINUE
20 CONTINUE
30 CONTINUE
JZ = J1
JU = 0
C
C GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING
C
NM1 = N - 1
IF (NM1 .LT. 1) GO TO 130
DO 120 K = 1, NM1
KP1 = K + 1
C
C ZERO NEXT FILL-IN COLUMN
C
JZ = JZ + 1
IF (JZ .GT. N) GO TO 50
IF (ML .LT. 1) GO TO 50
DO 40 I = 1, ML
ABD(I,JZ) = 0.0E0
40 CONTINUE
50 CONTINUE
C
C FIND L = PIVOT INDEX
C
LM = MIN(ML,N-K)
L = ISAMAX(LM+1,ABD(M,K),1) + M - 1
IPVT(K) = L + K - M
C
C ZERO PIVOT IMPLIES THIS COLUMN ALREADY TRIANGULARIZED
C
IF (ABD(L,K) .EQ. 0.0E0) GO TO 100
C
C INTERCHANGE IF NECESSARY
C
IF (L .EQ. M) GO TO 60
T = ABD(L,K)
ABD(L,K) = ABD(M,K)
ABD(M,K) = T
60 CONTINUE
C
C COMPUTE MULTIPLIERS
C
T = -1.0E0/ABD(M,K)
CALL SSCAL(LM,T,ABD(M+1,K),1)
C
C ROW ELIMINATION WITH COLUMN INDEXING
C
JU = MIN(MAX(JU,MU+IPVT(K)),N)
MM = M
IF (JU .LT. KP1) GO TO 90
DO 80 J = KP1, JU
L = L - 1
MM = MM - 1
T = ABD(L,J)
IF (L .EQ. MM) GO TO 70
ABD(L,J) = ABD(MM,J)
ABD(MM,J) = T
70 CONTINUE
CALL SAXPY(LM,T,ABD(M+1,K),1,ABD(MM+1,J),1)
80 CONTINUE
90 CONTINUE
GO TO 110
100 CONTINUE
INFO = K
110 CONTINUE
120 CONTINUE
130 CONTINUE
IPVT(N) = N
IF (ABD(M,N) .EQ. 0.0E0) INFO = N
RETURN
END