SUBROUTINE SGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z) INTEGER LDA,N,ML,MU,IPVT(1) REAL ABD(LDA,1),Z(1) REAL RCOND C C SGBCO FACTORS A REAL BAND MATRIX BY GAUSSIAN C ELIMINATION AND ESTIMATES THE CONDITION OF THE MATRIX. C C IF RCOND IS NOT NEEDED, SGBFA IS SLIGHTLY FASTER. C TO SOLVE A*X = B , FOLLOW SGBCO BY SGBSL. C TO COMPUTE INVERSE(A)*C , FOLLOW SGBCO BY SGBSL. C TO COMPUTE DETERMINANT(A) , FOLLOW SGBCO BY SGBDI. 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 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 RCOND REAL C AN ESTIMATE OF THE RECIPROCAL CONDITION OF A . C FOR THE SYSTEM A*X = B , RELATIVE PERTURBATIONS C IN A AND B OF SIZE EPSILON MAY CAUSE C RELATIVE PERTURBATIONS IN X OF SIZE EPSILON/RCOND . C IF RCOND IS SO SMALL THAT THE LOGICAL EXPRESSION C 1.0 + RCOND .EQ. 1.0 C IS TRUE, THEN A MAY BE SINGULAR TO WORKING C PRECISION. IN PARTICULAR, RCOND IS ZERO IF C EXACT SINGULARITY IS DETECTED OR THE ESTIMATE C UNDERFLOWS. C C Z REAL(N) C A WORK VECTOR WHOSE CONTENTS ARE USUALLY UNIMPORTANT. C IF A IS CLOSE TO A SINGULAR MATRIX, THEN Z IS C AN APPROXIMATE NULL VECTOR IN THE SENSE THAT C NORM(A*Z) = RCOND*NORM(A)*NORM(Z) . 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 = MAX0(1, J-MU) C I2 = MIN0(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 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 ABD SHOULD CONTAIN C C * * * + + + , * = NOT USED C * * 13 24 35 46 , + = USED FOR PIVOTING C * 12 23 34 45 56 C 11 22 33 44 55 66 C 21 32 43 54 65 * C C LINPACK. THIS VERSION DATED 08/14/78 . C CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB. C C SUBROUTINES AND FUNCTIONS C C LINPACK SGBFA C BLAS SAXPY,SDOT,SSCAL,SASUM C FORTRAN ABS,AMAX1,MAX0,MIN0,SIGN C C INTERNAL VARIABLES C REAL SDOT,EK,T,WK,WKM REAL ANORM,S,SASUM,SM,YNORM INTEGER IS,INFO,J,JU,K,KB,KP1,L,LA,LM,LZ,M,MM C C C COMPUTE 1-NORM OF A C ANORM = 0.0E0 L = ML + 1 IS = L + MU DO 10 J = 1, N ANORM = AMAX1(ANORM,SASUM(L,ABD(IS,J),1)) IF (IS .GT. ML + 1) IS = IS - 1 IF (J .LE. MU) L = L + 1 IF (J .GE. N - ML) L = L - 1 10 CONTINUE C C FACTOR C CALL SGBFA(ABD,LDA,N,ML,MU,IPVT,INFO) C C RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) . C ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND TRANS(A)*Y = E . C TRANS(A) IS THE TRANSPOSE OF A . THE COMPONENTS OF E ARE C CHOSEN TO CAUSE MAXIMUM LOCAL GROWTH IN THE ELEMENTS OF W WHERE C TRANS(U)*W = E . THE VECTORS ARE FREQUENTLY RESCALED TO AVOID C OVERFLOW. C C SOLVE TRANS(U)*W = E C EK = 1.0E0 DO 20 J = 1, N Z(J) = 0.0E0 20 CONTINUE M = ML + MU + 1 JU = 0 DO 100 K = 1, N IF (Z(K) .NE. 0.0E0) EK = SIGN(EK,-Z(K)) IF (ABS(EK-Z(K)) .LE. ABS(ABD(M,K))) GO TO 30 S = ABS(ABD(M,K))/ABS(EK-Z(K)) CALL SSCAL(N,S,Z,1) EK = S*EK 30 CONTINUE WK = EK - Z(K) WKM = -EK - Z(K) S = ABS(WK) SM = ABS(WKM) IF (ABD(M,K) .EQ. 0.0E0) GO TO 40 WK = WK/ABD(M,K) WKM = WKM/ABD(M,K) GO TO 50 40 CONTINUE WK = 1.0E0 WKM = 1.0E0 50 CONTINUE KP1 = K + 1 JU = MIN0(MAX0(JU,MU+IPVT(K)),N) MM = M IF (KP1 .GT. JU) GO TO 90 DO 60 J = KP1, JU MM = MM - 1 SM = SM + ABS(Z(J)+WKM*ABD(MM,J)) Z(J) = Z(J) + WK*ABD(MM,J) S = S + ABS(Z(J)) 60 CONTINUE IF (S .GE. SM) GO TO 80 T = WKM - WK WK = WKM MM = M DO 70 J = KP1, JU MM = MM - 1 Z(J) = Z(J) + T*ABD(MM,J) 70 CONTINUE 80 CONTINUE 90 CONTINUE Z(K) = WK 100 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) C C SOLVE TRANS(L)*Y = W C DO 120 KB = 1, N K = N + 1 - KB LM = MIN0(ML,N-K) IF (K .LT. N) Z(K) = Z(K) + SDOT(LM,ABD(M+1,K),1,Z(K+1),1) IF (ABS(Z(K)) .LE. 1.0E0) GO TO 110 S = 1.0E0/ABS(Z(K)) CALL SSCAL(N,S,Z,1) 110 CONTINUE L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T 120 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) C YNORM = 1.0E0 C C SOLVE L*V = Y C DO 140 K = 1, N L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T LM = MIN0(ML,N-K) IF (K .LT. N) CALL SAXPY(LM,T,ABD(M+1,K),1,Z(K+1),1) IF (ABS(Z(K)) .LE. 1.0E0) GO TO 130 S = 1.0E0/ABS(Z(K)) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM 130 CONTINUE 140 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM C C SOLVE U*Z = W C DO 160 KB = 1, N K = N + 1 - KB IF (ABS(Z(K)) .LE. ABS(ABD(M,K))) GO TO 150 S = ABS(ABD(M,K))/ABS(Z(K)) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM 150 CONTINUE IF (ABD(M,K) .NE. 0.0E0) Z(K) = Z(K)/ABD(M,K) IF (ABD(M,K) .EQ. 0.0E0) Z(K) = 1.0E0 LM = MIN0(K,M) - 1 LA = M - LM LZ = K - LM T = -Z(K) CALL SAXPY(LM,T,ABD(LA,K),1,Z(LZ),1) 160 CONTINUE C MAKE ZNORM = 1.0 S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM C IF (ANORM .NE. 0.0E0) RCOND = YNORM/ANORM IF (ANORM .EQ. 0.0E0) RCOND = 0.0E0 RETURN END