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C ALGORITHM 618, COLLECTED ALGORITHMS FROM ACM.
C ALGORITHM APPEARED IN ACM-TRANS. MATH. SOFTWARE, VOL.10, NO. 3,
C SEP., 1984, P. 346-347.
C$$$ PROGRAM TEST
C$$$C **********
C$$$C
C$$$C THIS IS A TEST PROGRAM FOR SUBROUTINES DSM AND FDJS.
C$$$C THE TEST DATA REPRESENTS A NEUTRON KINETICS PROBLEM.
C$$$C
C$$$C **********
C$$$ INTEGER I,INFO,IP,J,JP,L,LIWA,M,
C$$$ * MAXGRP,MAXROW,MINGRP,MINROW,N,NNZ,NUMGRP,NWRITE
C$$$ INTEGER INDCOL(6000),INDROW(6000),IPNTR(1201),JPNTR(1201),
C$$$ * NGRP(1200),IWA(7200)
C$$$ LOGICAL COL
C$$$ REAL DNSM,ERRIJ,ERRMAX,FJACT,FJACTR,SUM
C$$$ REAL D(1200),FJAC(6000),FJACD(1200),FVEC(1200),X(1200),XD(1200)
C$$$C
C$$$C LOGICAL OUTPUT UNIT IS ASSUMED TO BE NUMBER 6.
C$$$C
C$$$ DATA NWRITE /6/
C$$$C
C$$$ LIWA = 7200
C$$$ COL = .TRUE.
C$$$C
C$$$C TEST FOR DSM AND FDJS.
C$$$C
C$$$ WRITE (NWRITE,1000)
C$$$ DO 130 N = 300, 1200, 300
C$$$ WRITE (NWRITE,2000)
C$$$C
C$$$C DEFINITION OF SPARSITY PATTERN.
C$$$C
C$$$ M = N
C$$$ L = N/3
C$$$ NNZ = 0
C$$$ DO 10 J = 1, N
C$$$ NNZ = NNZ + 1
C$$$ INDROW(NNZ) = J
C$$$ INDCOL(NNZ) = J
C$$$ IF (MOD(J,L) .NE. 0) THEN
C$$$ NNZ = NNZ + 1
C$$$ INDROW(NNZ) = J + 1
C$$$ INDCOL(NNZ) = J
C$$$ END IF
C$$$ IF (J .LE. 2*L) THEN
C$$$ NNZ = NNZ + 1
C$$$ INDROW(NNZ) = J + L
C$$$ INDCOL(NNZ) = J
C$$$ IF (MOD(J,L) .NE. 1) THEN
C$$$ NNZ = NNZ + 1
C$$$ INDROW(NNZ) = J - 1
C$$$ INDCOL(NNZ) = J
C$$$ END IF
C$$$ END IF
C$$$ NNZ = NNZ + 1
C$$$ IF (J .GT. L) THEN
C$$$ INDROW(NNZ) = J - L
C$$$ ELSE
C$$$ INDROW(NNZ) = J + 2*L
C$$$ END IF
C$$$ INDCOL(NNZ) = J
C$$$ 10 CONTINUE
C$$$C
C$$$C CALL DSM.
C$$$C
C$$$ CALL DSM(M,N,NNZ,INDROW,INDCOL,NGRP,MAXGRP,MINGRP,
C$$$ * INFO,IPNTR,JPNTR,IWA,LIWA)
C$$$ IF (INFO .LE. 0) WRITE (NWRITE,4000) INFO
C$$$C
C$$$C STATISTICS FOR THE MATRIX.
C$$$C
C$$$ MAXROW = 0
C$$$ MINROW = N
C$$$ DO 20 I = 1, M
C$$$ MAXROW = MAX(MAXROW,IPNTR(I+1)-IPNTR(I))
C$$$ MINROW = MIN(MINROW,IPNTR(I+1)-IPNTR(I))
C$$$ 20 CONTINUE
C$$$ DNSM = FLOAT(100*NNZ)/FLOAT(M*N)
C$$$ WRITE (NWRITE,3000) N,NNZ,DNSM,MINROW,MAXROW,MINGRP,MAXGRP
C$$$C
C$$$C TEST FOR FDJS.
C$$$C
C$$$ DO 30 J = 1, N
C$$$ X(J) = FLOAT(J)/FLOAT(N)
C$$$ 30 CONTINUE
C$$$ CALL FCN(N,X,INDCOL,IPNTR,FVEC)
C$$$C
C$$$C APPROXIMATE THE JACOBIAN MATRIX.
C$$$C
C$$$ DO 60 NUMGRP = 1, MAXGRP
C$$$ DO 40 J = 1, N
C$$$ D(J) = 0.0
C$$$ IF (NGRP(J) .EQ. NUMGRP) D(J) = 0.001
C$$$ XD(J) = X(J) + D(J)
C$$$ 40 CONTINUE
C$$$ CALL FCN(N,XD,INDCOL,IPNTR,FJACD)
C$$$ DO 50 I = 1, M
C$$$ FJACD(I) = FJACD(I) - FVEC(I)
C$$$ 50 CONTINUE
C$$$ IF (COL) THEN
C$$$ CALL FDJS(M,N,COL,INDROW,JPNTR,NGRP,NUMGRP,D,FJACD,FJAC)
C$$$ ELSE
C$$$ CALL FDJS(M,N,COL,INDCOL,IPNTR,NGRP,NUMGRP,D,FJACD,FJAC)
C$$$ END IF
C$$$ 60 CONTINUE
C$$$C
C$$$C TEST THE APPROXIMATION TO THE JACOBIAN.
C$$$C
C$$$ ERRMAX = 0.0
C$$$ IF (COL) THEN
C$$$C
C$$$C TEST FOR THE COLUMN-ORIENTED DEFINITION OF
C$$$C THE SPARSITY PATTERN.
C$$$C
C$$$ DO 90 J = 1, N
C$$$ DO 80 JP = JPNTR(J), JPNTR(J+1)-1
C$$$ I = INDROW(JP)
C$$$ SUM = 0.0
C$$$ DO 70 IP = IPNTR(I), IPNTR(I+1)-1
C$$$ SUM = SUM + X(INDCOL(IP))
C$$$ 70 CONTINUE
C$$$ SUM = SUM + X(I)
C$$$ FJACT = 1.0 + 2.0*SUM
C$$$ IF (I .EQ. J) FJACT = 2.0*FJACT
C$$$ ERRIJ = FJAC(JP) - FJACT
C$$$ IF (FJACT .NE. 0.0) ERRIJ = ERRIJ/FJACT
C$$$ ERRMAX = MAX(ERRMAX,ABS(ERRIJ))
C$$$ 80 CONTINUE
C$$$ 90 CONTINUE
C$$$ ELSE
C$$$C
C$$$C TEST FOR THE ROW-ORIENTED DEFINITION OF
C$$$C THE SPARSITY PATTERN.
C$$$C
C$$$ DO 120 I = 1, M
C$$$ SUM = 0.0
C$$$ DO 100 IP = IPNTR(I), IPNTR(I+1)-1
C$$$ SUM = SUM + X(INDCOL(IP))
C$$$ 100 CONTINUE
C$$$ SUM = SUM + X(I)
C$$$ FJACTR = 1.0 + 2.0*SUM
C$$$ DO 110 IP = IPNTR(I), IPNTR(I+1)-1
C$$$ J = INDCOL(IP)
C$$$ FJACT = FJACTR
C$$$ IF (I .EQ. J) FJACT = 2.0*FJACT
C$$$ ERRIJ = FJAC(IP) - FJACT
C$$$ IF (FJACT .NE. 0.0) ERRIJ = ERRIJ/FJACT
C$$$ ERRMAX = MAX(ERRMAX,ABS(ERRIJ))
C$$$ 110 CONTINUE
C$$$ 120 CONTINUE
C$$$ END IF
C$$$ WRITE (NWRITE,5000) ERRMAX
C$$$ COL = .NOT. COL
C$$$ 130 CONTINUE
C$$$ STOP
C$$$C
C$$$C FORMAT STATEMENTS.
C$$$C
C$$$ 1000 FORMAT(// ' TESTS FOR DSM AND FDJS - NEUTRON KINETICS PROBLEM'//
C$$$ * ' STATISTICS GENERATED '//
C$$$ * ' N - NUMBER OF COLUMNS '/
C$$$ * ' NNZ - NUMBER OF NON-ZERO ELEMENTS'/
C$$$ * ' DNSM - MATRIX DENSITY (PERCENTAGE)'/
C$$$ * ' MINROW - MINIMUM NUMBER OF NON-ZEROS IN ANY ROW'/
C$$$ * ' MAXROW - MAXIMUM NUMBER OF NON-ZEROS IN ANY ROW'/
C$$$ * ' MINGRP - LOWER BOUND ON NUMBER OF GROUPS'/
C$$$ * ' MAXGRP - NUMBER OF GROUPS DETERMINED BY DSM'//)
C$$$ 2000 FORMAT(// 3X,'N',6X,'NNZ',5X,'DNSM',5X,
C$$$ * 'MINROW',4X,'MAXROW',4X,'MINGRP',4X,'MAXGRP'//)
C$$$ 3000 FORMAT(2(I5,3X),F6.2,4X,4(I5,5X))
C$$$ 4000 FORMAT(// ' *** MISTAKE IN INPUT DATA, INFO IS ***',I6)
C$$$ 5000 FORMAT(// ' LARGEST RELATIVE ERROR OF APPROXIMATION IS',E10.2)
C$$$ END
C$$$ SUBROUTINE FCN(N,X,INDCOL,IPNTR,FVEC)
C$$$ INTEGER N
C$$$ INTEGER INDCOL(*),IPNTR(N+1)
C$$$ REAL X(N),FVEC(N)
C$$$C
C$$$C FUNCTION SUBROUTINE FOR TESTING FDJS.
C$$$C
C$$$ INTEGER I,IP
C$$$ REAL SUM
C$$$ DO 20 I = 1, N
C$$$ SUM = 0.0
C$$$ DO 10 IP = IPNTR(I), IPNTR(I+1)-1
C$$$ SUM = SUM + X(INDCOL(IP))
C$$$ 10 CONTINUE
C$$$ SUM = SUM + X(I)
C$$$ FVEC(I) = SUM*(1.0 + SUM) + 1.0
C$$$ 20 CONTINUE
C$$$ RETURN
C$$$C
C$$$C LAST CARD OF SUBROUTINE FCN.
C$$$C
C$$$ END
SUBROUTINE DSM(M,N,NPAIRS,INDROW,INDCOL,NGRP,MAXGRP,MINGRP,
* INFO,IPNTR,JPNTR,IWA,LIWA)
INTEGER M,N,NPAIRS,MAXGRP,MINGRP,INFO,LIWA
INTEGER INDROW(NPAIRS),INDCOL(NPAIRS),NGRP(N),
* IPNTR(M+1),JPNTR(N+1),IWA(LIWA)
C **********
C
C SUBROUTINE DSM
C
C THE PURPOSE OF DSM IS TO DETERMINE AN OPTIMAL OR NEAR-
C OPTIMAL CONSISTENT PARTITION OF THE COLUMNS OF A SPARSE
C M BY N MATRIX A.
C
C THE SPARSITY PATTERN OF THE MATRIX A IS SPECIFIED BY
C THE ARRAYS INDROW AND INDCOL. ON INPUT THE INDICES
C FOR THE NON-ZERO ELEMENTS OF A ARE
C
C INDROW(K),INDCOL(K), K = 1,2,...,NPAIRS.
C
C THE (INDROW,INDCOL) PAIRS MAY BE SPECIFIED IN ANY ORDER.
C DUPLICATE INPUT PAIRS ARE PERMITTED, BUT THE SUBROUTINE
C ELIMINATES THEM.
C
C THE SUBROUTINE PARTITIONS THE COLUMNS OF A INTO GROUPS
C SUCH THAT COLUMNS IN THE SAME GROUP DO NOT HAVE A
C NON-ZERO IN THE SAME ROW POSITION. A PARTITION OF THE
C COLUMNS OF A WITH THIS PROPERTY IS CONSISTENT WITH THE
C DIRECT DETERMINATION OF A.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE DSM(M,N,NPAIRS,INDROW,INDCOL,NGRP,MAXGRP,MINGRP,
C INFO,IPNTR,JPNTR,IWA,LIWA)
C
C WHERE
C
C M IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER
C OF ROWS OF A.
C
C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER
C OF COLUMNS OF A.
C
C NPAIRS IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE
C NUMBER OF (INDROW,INDCOL) PAIRS USED TO DESCRIBE THE
C SPARSITY PATTERN OF A.
C
C INDROW IS AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT INDROW
C MUST CONTAIN THE ROW INDICES OF THE NON-ZERO ELEMENTS OF A.
C ON OUTPUT INDROW IS PERMUTED SO THAT THE CORRESPONDING
C COLUMN INDICES ARE IN NON-DECREASING ORDER. THE COLUMN
C INDICES CAN BE RECOVERED FROM THE ARRAY JPNTR.
C
C INDCOL IS AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT INDCOL
C MUST CONTAIN THE COLUMN INDICES OF THE NON-ZERO ELEMENTS OF
C A. ON OUTPUT INDCOL IS PERMUTED SO THAT THE CORRESPONDING
C ROW INDICES ARE IN NON-DECREASING ORDER. THE ROW INDICES
C CAN BE RECOVERED FROM THE ARRAY IPNTR.
C
C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES
C THE PARTITION OF THE COLUMNS OF A. COLUMN JCOL BELONGS
C TO GROUP NGRP(JCOL).
C
C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE
C NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF A.
C
C MINGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES A LOWER
C BOUND FOR THE NUMBER OF GROUPS IN ANY CONSISTENT PARTITION
C OF THE COLUMNS OF A.
C
C INFO IS AN INTEGER OUTPUT VARIABLE SET AS FOLLOWS. FOR
C NORMAL TERMINATION INFO = 1. IF M, N, OR NPAIRS IS NOT
C POSITIVE OR LIWA IS LESS THAN MAX(M,6*N), THEN INFO = 0.
C IF THE K-TH ELEMENT OF INDROW IS NOT AN INTEGER BETWEEN
C 1 AND M OR THE K-TH ELEMENT OF INDCOL IS NOT AN INTEGER
C BETWEEN 1 AND N, THEN INFO = -K.
C
C IPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH M + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL.
C THE COLUMN INDICES FOR ROW I ARE
C
C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1.
C
C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C JPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW.
C THE ROW INDICES FOR COLUMN J ARE
C
C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1.
C
C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C IWA IS AN INTEGER WORK ARRAY OF LENGTH LIWA.
C
C LIWA IS A POSITIVE INTEGER INPUT VARIABLE NOT LESS THAN
C MAX(M,6*N).
C
C SUBPROGRAMS CALLED
C
C MINPACK-SUPPLIED ... DEGR,IDO,NUMSRT,SEQ,SETR,SLO,SRTDAT
C
C FORTRAN-SUPPLIED ... MAX
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983.
C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE'
C
C **********
INTEGER I,IR,J,JP,K,MAXCLQ,NNZ,NUMGRP
C
C CHECK THE INPUT DATA.
C
INFO = 0
IF (M .LT. 1 .OR. N .LT. 1 .OR. NPAIRS .LT. 1 .OR.
* LIWA .LT. MAX(M,6*N)) RETURN
DO 10 K = 1, NPAIRS
INFO = -K
IF (INDROW(K) .LT. 1 .OR. INDROW(K) .GT. M .OR.
* INDCOL(K) .LT. 1 .OR. INDCOL(K) .GT. N) RETURN
10 CONTINUE
INFO = 1
C
C SORT THE DATA STRUCTURE BY COLUMNS.
C
CALL SRTDAT(N,NPAIRS,INDROW,INDCOL,JPNTR,IWA)
C
C COMPRESS THE DATA AND DETERMINE THE NUMBER OF
C NON-ZERO ELEMENTS OF A.
C
DO 20 I = 1, M
IWA(I) = 0
20 CONTINUE
NNZ = 1
DO 40 J = 1, N
K = NNZ
DO 30 JP = JPNTR(J), JPNTR(J+1)-1
IR = INDROW(JP)
IF (IWA(IR) .NE. J) THEN
INDROW(NNZ) = IR
NNZ = NNZ + 1
IWA(IR) = J
END IF
30 CONTINUE
JPNTR(J) = K
40 CONTINUE
JPNTR(N+1) = NNZ
C
C EXTEND THE DATA STRUCTURE TO ROWS.
C
CALL SETR(M,N,INDROW,JPNTR,INDCOL,IPNTR,IWA)
C
C DETERMINE A LOWER BOUND FOR THE NUMBER OF GROUPS.
C
MINGRP = 0
DO 50 I = 1, M
MINGRP = MAX(MINGRP,IPNTR(I+1)-IPNTR(I))
50 CONTINUE
C
C DETERMINE THE DEGREE SEQUENCE FOR THE INTERSECTION
C GRAPH OF THE COLUMNS OF A.
C
CALL DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(N+1))
C
C COLOR THE INTERSECTION GRAPH OF THE COLUMNS OF A
C WITH THE SMALLEST-LAST (SL) ORDERING.
C
CALL SLO(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(4*N+1),
* MAXCLQ,IWA(1),IWA(N+1),IWA(2*N+1),IWA(3*N+1))
CALL SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(4*N+1),NGRP,MAXGRP,
* IWA(N+1))
MINGRP = MAX(MINGRP,MAXCLQ)
C
C EXIT IF THE SMALLEST-LAST ORDERING IS OPTIMAL.
C
IF (MAXGRP .EQ. MINGRP) RETURN
C
C COLOR THE INTERSECTION GRAPH OF THE COLUMNS OF A
C WITH THE INCIDENCE-DEGREE (ID) ORDERING.
C
CALL IDO(M,N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(4*N+1),
* MAXCLQ,IWA(1),IWA(N+1),IWA(2*N+1),IWA(3*N+1))
CALL SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(4*N+1),IWA(1),NUMGRP,
* IWA(N+1))
MINGRP = MAX(MINGRP,MAXCLQ)
C
C RETAIN THE BETTER OF THE TWO ORDERINGS SO FAR.
C
IF (NUMGRP .LT. MAXGRP) THEN
MAXGRP = NUMGRP
DO 60 J = 1, N
NGRP(J) = IWA(J)
60 CONTINUE
C
C EXIT IF THE INCIDENCE-DEGREE ORDERING IS OPTIMAL.
C
IF (MAXGRP .EQ. MINGRP) RETURN
END IF
C
C COLOR THE INTERSECTION GRAPH OF THE COLUMNS OF A
C WITH THE LARGEST-FIRST (LF) ORDERING.
C
CALL NUMSRT(N,N-1,IWA(5*N+1),-1,IWA(4*N+1),IWA(2*N+1),IWA(N+1))
CALL SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(4*N+1),IWA(1),NUMGRP,
* IWA(N+1))
C
C RETAIN THE BEST OF THE THREE ORDERINGS AND EXIT.
C
IF (NUMGRP .LT. MAXGRP) THEN
MAXGRP = NUMGRP
DO 70 J = 1, N
NGRP(J) = IWA(J)
70 CONTINUE
END IF
RETURN
C
C LAST CARD OF SUBROUTINE DSM.
C
END
SUBROUTINE DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,IWA)
INTEGER N
INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),NDEG(N),IWA(N)
C **********
C
C SUBROUTINE DEGR
C
C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A,
C THIS SUBROUTINE DETERMINES THE DEGREE SEQUENCE FOR
C THE INTERSECTION GRAPH OF THE COLUMNS OF A.
C
C IN GRAPH-THEORY TERMINOLOGY, THE INTERSECTION GRAPH OF
C THE COLUMNS OF A IS THE LOOPLESS GRAPH G WITH VERTICES
C A(J), J = 1,2,...,N WHERE A(J) IS THE J-TH COLUMN OF A
C AND WITH EDGE (A(I),A(J)) IF AND ONLY IF COLUMNS I AND J
C HAVE A NON-ZERO IN THE SAME ROW POSITION.
C
C NOTE THAT THE VALUE OF M IS NOT NEEDED BY DEGR AND IS
C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,IWA)
C
C WHERE
C
C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER
C OF COLUMNS OF A.
C
C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW
C INDICES FOR THE NON-ZEROES IN THE MATRIX A.
C
C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW.
C THE ROW INDICES FOR COLUMN J ARE
C
C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1.
C
C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE
C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A.
C
C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL.
C THE COLUMN INDICES FOR ROW I ARE
C
C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1.
C
C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C NDEG IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH
C SPECIFIES THE DEGREE SEQUENCE. THE DEGREE OF THE
C J-TH COLUMN OF A IS NDEG(J).
C
C IWA IS AN INTEGER WORK ARRAY OF LENGTH N.
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983.
C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE'
C
C **********
INTEGER IC,IP,IR,JCOL,JP
C
C INITIALIZATION BLOCK.
C
DO 10 JP = 1, N
NDEG(JP) = 0
IWA(JP) = 0
10 CONTINUE
C
C COMPUTE THE DEGREE SEQUENCE BY DETERMINING THE CONTRIBUTIONS
C TO THE DEGREES FROM THE CURRENT(JCOL) COLUMN AND FURTHER
C COLUMNS WHICH HAVE NOT YET BEEN CONSIDERED.
C
DO 40 JCOL = 2, N
IWA(JCOL) = N
C
C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND
C TO NON-ZEROES IN THE MATRIX.
C
DO 30 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1
IR = INDROW(JP)
C
C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC)
C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX.
C
DO 20 IP = IPNTR(IR), IPNTR(IR+1)-1
IC = INDCOL(IP)
C
C ARRAY IWA MARKS COLUMNS WHICH HAVE CONTRIBUTED TO
C THE DEGREE COUNT OF COLUMN JCOL. UPDATE THE DEGREE
C COUNTS OF THESE COLUMNS AS WELL AS COLUMN JCOL.
C
IF (IWA(IC) .LT. JCOL) THEN
IWA(IC) = JCOL
NDEG(IC) = NDEG(IC) + 1
NDEG(JCOL) = NDEG(JCOL) + 1
END IF
20 CONTINUE
30 CONTINUE
40 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE DEGR.
C
END
SUBROUTINE IDO(M,N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST,
* MAXCLQ,IWA1,IWA2,IWA3,IWA4)
INTEGER M,N,MAXCLQ
INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(M+1),NDEG(N),
* LIST(N),IWA1(0:N-1),IWA2(N),IWA3(N),IWA4(N)
C **********
C
C SUBROUTINE IDO
C
C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS
C SUBROUTINE DETERMINES AN INCIDENCE-DEGREE ORDERING OF THE
C COLUMNS OF A.
C
C THE INCIDENCE-DEGREE ORDERING IS DEFINED FOR THE LOOPLESS
C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE
C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF
C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION.
C
C THE INCIDENCE-DEGREE ORDERING IS DETERMINED RECURSIVELY BY
C LETTING LIST(K), K = 1,...,N BE A COLUMN WITH MAXIMAL
C INCIDENCE TO THE SUBGRAPH SPANNED BY THE ORDERED COLUMNS.
C AMONG ALL THE COLUMNS OF MAXIMAL INCIDENCE, IDO CHOOSES A
C COLUMN OF MAXIMAL DEGREE.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE IDO(M,N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST,
C MAXCLQ,IWA1,IWA2,IWA3,IWA4)
C
C WHERE
C
C M IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER
C OF ROWS OF A.
C
C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER
C OF COLUMNS OF A.
C
C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW
C INDICES FOR THE NON-ZEROES IN THE MATRIX A.
C
C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW.
C THE ROW INDICES FOR COLUMN J ARE
C
C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1.
C
C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE
C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A.
C
C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL.
C THE COLUMN INDICES FOR ROW I ARE
C
C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1.
C
C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C NDEG IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES
C THE DEGREE SEQUENCE. THE DEGREE OF THE J-TH COLUMN
C OF A IS NDEG(J).
C
C LIST IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES
C THE INCIDENCE-DEGREE ORDERING OF THE COLUMNS OF A. THE J-TH
C COLUMN IN THIS ORDER IS LIST(J).
C
C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE
C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING.
C
C IWA1,IWA2,IWA3, AND IWA4 ARE INTEGER WORK ARRAYS OF LENGTH N.
C
C SUBPROGRAMS CALLED
C
C MINPACK-SUPPLIED ... NUMSRT
C
C FORTRAN-SUPPLIED ... MAX
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983.
C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE'
C
C **********
INTEGER IC,IP,IR,JCOL,JP,
* MAXINC,MAXLST,NCOMP,NUMINC,NUMLST,NUMORD,NUMWGT
C
C SORT THE DEGREE SEQUENCE.
C
CALL NUMSRT(N,N-1,NDEG,-1,IWA4,IWA2,IWA3)
C
C INITIALIZATION BLOCK.
C
C CREATE A DOUBLY-LINKED LIST TO ACCESS THE INCIDENCES OF THE
C COLUMNS. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS.
C
C EACH UN-ORDERED COLUMN IC IS IN A LIST (THE INCIDENCE LIST)
C OF COLUMNS WITH THE SAME INCIDENCE.
C
C IWA1(NUMINC) IS THE FIRST COLUMN IN THE NUMINC LIST
C UNLESS IWA1(NUMINC) = 0. IN THIS CASE THERE ARE
C NO COLUMNS IN THE NUMINC LIST.
C
C IWA2(IC) IS THE COLUMN BEFORE IC IN THE INCIDENCE LIST
C UNLESS IWA2(IC) = 0. IN THIS CASE IC IS THE FIRST
C COLUMN IN THIS INCIDENCE LIST.
C
C IWA3(IC) IS THE COLUMN AFTER IC IN THE INCIDENCE LIST
C UNLESS IWA3(IC) = 0. IN THIS CASE IC IS THE LAST
C COLUMN IN THIS INCIDENCE LIST.
C
C IF IC IS AN UN-ORDERED COLUMN, THEN LIST(IC) IS THE
C INCIDENCE OF IC TO THE GRAPH INDUCED BY THE ORDERED
C COLUMNS. IF JCOL IS AN ORDERED COLUMN, THEN LIST(JCOL)
C IS THE INCIDENCE-DEGREE ORDER OF COLUMN JCOL.
C
MAXINC = 0
DO 10 JP = N, 1, -1
IC = IWA4(JP)
IWA1(N-JP) = 0
IWA2(IC) = 0
IWA3(IC) = IWA1(0)
IF (IWA1(0) .GT. 0) IWA2(IWA1(0)) = IC
IWA1(0) = IC
IWA4(JP) = 0
LIST(JP) = 0
10 CONTINUE
C
C DETERMINE THE MAXIMAL SEARCH LENGTH FOR THE LIST
C OF COLUMNS OF MAXIMAL INCIDENCE.
C
MAXLST = 0
DO 20 IR = 1, M
MAXLST = MAXLST + (IPNTR(IR+1) - IPNTR(IR))**2
20 CONTINUE
MAXLST = MAXLST/N
MAXCLQ = 0
NUMORD = 1
C
C BEGINNING OF ITERATION LOOP.
C
30 CONTINUE
C
C UPDATE THE SIZE OF THE LARGEST CLIQUE
C FOUND DURING THE ORDERING.
C
IF (MAXINC .EQ. 0) NCOMP = 0
NCOMP = NCOMP + 1
IF (MAXINC + 1 .EQ. NCOMP) MAXCLQ = MAX(MAXCLQ,NCOMP)
C
C CHOOSE A COLUMN JCOL OF MAXIMAL DEGREE AMONG THE
C COLUMNS OF MAXIMAL INCIDENCE MAXINC.
C
40 CONTINUE
JP = IWA1(MAXINC)
IF (JP .GT. 0) GO TO 50
MAXINC = MAXINC - 1
GO TO 40
50 CONTINUE
NUMWGT = -1
DO 60 NUMLST = 1, MAXLST
IF (NDEG(JP) .GT. NUMWGT) THEN
NUMWGT = NDEG(JP)
JCOL = JP
END IF
JP = IWA3(JP)
IF (JP .LE. 0) GO TO 70
60 CONTINUE
70 CONTINUE
LIST(JCOL) = NUMORD
NUMORD = NUMORD + 1
C
C TERMINATION TEST.
C
IF (NUMORD .GT. N) GO TO 100
C
C DELETE COLUMN JCOL FROM THE MAXINC LIST.
C
IF (IWA2(JCOL) .EQ. 0) THEN
IWA1(MAXINC) = IWA3(JCOL)
ELSE
IWA3(IWA2(JCOL)) = IWA3(JCOL)
END IF
IF (IWA3(JCOL) .GT. 0) IWA2(IWA3(JCOL)) = IWA2(JCOL)
C
C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL.
C
IWA4(JCOL) = N
C
C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND
C TO NON-ZEROES IN THE MATRIX.
C
DO 90 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1
IR = INDROW(JP)
C
C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC)
C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX.
C
DO 80 IP = IPNTR(IR), IPNTR(IR+1)-1
IC = INDCOL(IP)
C
C ARRAY IWA4 MARKS COLUMNS WHICH ARE ADJACENT TO
C COLUMN JCOL.
C
IF (IWA4(IC) .LT. NUMORD) THEN
IWA4(IC) = NUMORD
C
C UPDATE THE POINTERS TO THE CURRENT INCIDENCE LISTS.
C
NUMINC = LIST(IC)
LIST(IC) = LIST(IC) + 1
MAXINC = MAX(MAXINC,LIST(IC))
C
C DELETE COLUMN IC FROM THE NUMINC LIST.
C
IF (IWA2(IC) .EQ. 0) THEN
IWA1(NUMINC) = IWA3(IC)
ELSE
IWA3(IWA2(IC)) = IWA3(IC)
END IF
IF (IWA3(IC) .GT. 0) IWA2(IWA3(IC)) = IWA2(IC)
C
C ADD COLUMN IC TO THE NUMINC+1 LIST.
C
IWA2(IC) = 0
IWA3(IC) = IWA1(NUMINC+1)
IF (IWA1(NUMINC+1) .GT. 0) IWA2(IWA1(NUMINC+1)) = IC
IWA1(NUMINC+1) = IC
END IF
80 CONTINUE
90 CONTINUE
C
C END OF ITERATION LOOP.
C
GO TO 30
100 CONTINUE
C
C INVERT THE ARRAY LIST.
C
DO 110 JCOL = 1, N
IWA2(LIST(JCOL)) = JCOL
110 CONTINUE
DO 120 JP = 1, N
LIST(JP) = IWA2(JP)
120 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE IDO.
C
END
SUBROUTINE NUMSRT(N,NMAX,NUM,MODE,INDEX,LAST,NEXT)
INTEGER N,NMAX,MODE
INTEGER NUM(N),INDEX(N),LAST(0:NMAX),NEXT(N)
C **********.
C
C SUBROUTINE NUMSRT
C
C GIVEN A SEQUENCE OF INTEGERS, THIS SUBROUTINE GROUPS
C TOGETHER THOSE INDICES WITH THE SAME SEQUENCE VALUE
C AND, OPTIONALLY, SORTS THE SEQUENCE INTO EITHER
C ASCENDING OR DESCENDING ORDER.
C
C THE SEQUENCE OF INTEGERS IS DEFINED BY THE ARRAY NUM,
C AND IT IS ASSUMED THAT THE INTEGERS ARE EACH FROM THE SET
C 0,1,...,NMAX. ON OUTPUT THE INDICES K SUCH THAT NUM(K) = L
C FOR ANY L = 0,1,...,NMAX CAN BE OBTAINED FROM THE ARRAYS
C LAST AND NEXT AS FOLLOWS.
C
C K = LAST(L)
C WHILE (K .NE. 0) K = NEXT(K)
C
C OPTIONALLY, THE SUBROUTINE PRODUCES AN ARRAY INDEX SO THAT
C THE SEQUENCE NUM(INDEX(I)), I = 1,2,...,N IS SORTED.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE NUMSRT(N,NMAX,NUM,MODE,INDEX,LAST,NEXT)
C
C WHERE
C
C N IS A POSITIVE INTEGER INPUT VARIABLE.
C
C NMAX IS A POSITIVE INTEGER INPUT VARIABLE.
C
C NUM IS AN INPUT ARRAY OF LENGTH N WHICH CONTAINS THE
C SEQUENCE OF INTEGERS TO BE GROUPED AND SORTED. IT
C IS ASSUMED THAT THE INTEGERS ARE EACH FROM THE SET
C 0,1,...,NMAX.
C
C MODE IS AN INTEGER INPUT VARIABLE. THE SEQUENCE NUM IS
C SORTED IN ASCENDING ORDER IF MODE IS POSITIVE AND IN
C DESCENDING ORDER IF MODE IS NEGATIVE. IF MODE IS 0,
C NO SORTING IS DONE.
C
C INDEX IS AN INTEGER OUTPUT ARRAY OF LENGTH N SET SO
C THAT THE SEQUENCE
C
C NUM(INDEX(I)), I = 1,2,...,N
C
C IS SORTED ACCORDING TO THE SETTING OF MODE. IF MODE
C IS 0, INDEX IS NOT REFERENCED.
C
C LAST IS AN INTEGER OUTPUT ARRAY OF LENGTH NMAX + 1. THE
C INDEX OF NUM FOR THE LAST OCCURRENCE OF L IS LAST(L)
C FOR ANY L = 0,1,...,NMAX UNLESS LAST(L) = 0. IN
C THIS CASE L DOES NOT APPEAR IN NUM.
C
C NEXT IS AN INTEGER OUTPUT ARRAY OF LENGTH N. IF
C NUM(K) = L, THEN THE INDEX OF NUM FOR THE PREVIOUS
C OCCURRENCE OF L IS NEXT(K) FOR ANY L = 0,1,...,NMAX
C UNLESS NEXT(K) = 0. IN THIS CASE THERE IS NO PREVIOUS
C OCCURRENCE OF L IN NUM.
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983.
C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE'
C
C **********
INTEGER I,J,JINC,JL,JU,K,L
C
C DETERMINE THE ARRAYS NEXT AND LAST.
C
DO 10 I = 0, NMAX
LAST(I) = 0
10 CONTINUE
DO 20 K = 1, N
L = NUM(K)
NEXT(K) = LAST(L)
LAST(L) = K
20 CONTINUE
IF (MODE .EQ. 0) RETURN
C
C STORE THE POINTERS TO THE SORTED ARRAY IN INDEX.
C
I = 1
IF (MODE .GT. 0) THEN
JL = 0
JU = NMAX
JINC = 1
ELSE
JL = NMAX
JU = 0
JINC = -1
END IF
DO 50 J = JL, JU, JINC
K = LAST(J)
30 CONTINUE
IF (K .EQ. 0) GO TO 40
INDEX(I) = K
I = I + 1
K = NEXT(K)
GO TO 30
40 CONTINUE
50 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE NUMSRT.
C
END
SUBROUTINE SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,LIST,NGRP,MAXGRP,
* IWA)
INTEGER N,MAXGRP
INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),LIST(N),
* NGRP(N),IWA(N)
C **********
C
C SUBROUTINE SEQ
C
C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS
C SUBROUTINE DETERMINES A CONSISTENT PARTITION OF THE
C COLUMNS OF A BY A SEQUENTIAL ALGORITHM.
C
C A CONSISTENT PARTITION IS DEFINED IN TERMS OF THE LOOPLESS
C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE
C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF
C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION.
C
C A PARTITION OF THE COLUMNS OF A INTO GROUPS IS CONSISTENT
C IF THE COLUMNS IN ANY GROUP ARE NOT ADJACENT IN THE GRAPH G.
C IN GRAPH-THEORY TERMINOLOGY, A CONSISTENT PARTITION OF THE
C COLUMNS OF A CORRESPONDS TO A COLORING OF THE GRAPH G.
C
C THE SUBROUTINE EXAMINES THE COLUMNS IN THE ORDER SPECIFIED
C BY THE ARRAY LIST, AND ASSIGNS THE CURRENT COLUMN TO THE
C GROUP WITH THE SMALLEST POSSIBLE NUMBER.
C
C NOTE THAT THE VALUE OF M IS NOT NEEDED BY SEQ AND IS
C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,LIST,NGRP,MAXGRP,
C IWA)
C
C WHERE
C
C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER
C OF COLUMNS OF A.
C
C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW
C INDICES FOR THE NON-ZEROES IN THE MATRIX A.
C
C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW.
C THE ROW INDICES FOR COLUMN J ARE
C
C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1.
C
C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE
C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A.
C
C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH
C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL.
C THE COLUMN INDICES FOR ROW I ARE
C
C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1.
C
C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO
C ELEMENTS OF THE MATRIX A.
C
C LIST IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES
C THE ORDER TO BE USED BY THE SEQUENTIAL ALGORITHM.
C THE J-TH COLUMN IN THIS ORDER IS LIST(J).
C
C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES
C THE PARTITION OF THE COLUMNS OF A. COLUMN JCOL BELONGS
C TO GROUP NGRP(JCOL).
C
C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE
C NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF A.
C
C IWA IS AN INTEGER WORK ARRAY OF LENGTH N.
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983.
C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE'
C
C **********
INTEGER IC,IP,IR,J,JCOL,JP
C
C INITIALIZATION BLOCK.
C
MAXGRP = 0
DO 10 JP = 1, N
NGRP(JP) = N
IWA(JP) = 0
10 CONTINUE
C
C BEGINNING OF ITERATION LOOP.
C
DO 60 J = 1, N
JCOL = LIST(J)
C
C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL.
C
C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND
C TO NON-ZEROES IN THE MATRIX.
C
DO 30 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1
IR = INDROW(JP)
C
C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC)
C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX.
C
DO 20 IP = IPNTR(IR), IPNTR(IR+1)-1
IC = INDCOL(IP)
C
C ARRAY IWA MARKS THE GROUP NUMBERS OF THE
C COLUMNS WHICH ARE ADJACENT TO COLUMN JCOL.
C
IWA(NGRP(IC)) = J
20 CONTINUE
30 CONTINUE
C
C ASSIGN THE SMALLEST UN-MARKED GROUP NUMBER TO JCOL.
C