SUBROUTINE DEC (N, NDIM, A, IP, IER)
C VERSION REAL DOUBLE PRECISION
INTEGER N,NDIM,IP,IER,NM1,K,KP1,M,I,J
DOUBLE PRECISION A,T
DIMENSION A(NDIM,N), IP(N)
C-----------------------------------------------------------------------
C MATRIX TRIANGULARIZATION BY GAUSSIAN ELIMINATION.
C INPUT..
C N = ORDER OF MATRIX.
C NDIM = DECLARED DIMENSION OF ARRAY A .
C A = MATRIX TO BE TRIANGULARIZED.
C OUTPUT..
C A(I,J), I.LE.J = UPPER TRIANGULAR FACTOR, U .
C A(I,J), I.GT.J = MULTIPLIERS = LOWER TRIANGULAR FACTOR, I - L.
C IP(K), K.LT.N = INDEX OF K-TH PIVOT ROW.
C IP(N) = (-1)**(NUMBER OF INTERCHANGES) OR O .
C IER = 0 IF MATRIX A IS NONSINGULAR, OR K IF FOUND TO BE
C SINGULAR AT STAGE K.
C USE SOL TO OBTAIN SOLUTION OF LINEAR SYSTEM.
C DETERM(A) = IP(N)*A(1,1)*A(2,2)*...*A(N,N).
C IF IP(N)=O, A IS SINGULAR, SOL WILL DIVIDE BY ZERO.
C
C REFERENCE..
C C. B. MOLER, ALGORITHM 423, LINEAR EQUATION SOLVER,
C C.A.C.M. 15 (1972), P. 274.
C-----------------------------------------------------------------------
IER = 0
IP(N) = 1
IF (N .EQ. 1) GO TO 70
NM1 = N - 1
DO 60 K = 1,NM1
KP1 = K + 1
M = K
DO 10 I = KP1,N
10 IF (DABS(A(I,K)) .GT. DABS(A(M,K))) M = I
IP(K) = M
T = A(M,K)
IF (M .EQ. K) GO TO 20
IP(N) = -IP(N)
A(M,K) = A(K,K)
A(K,K) = T
20 IF (T .EQ. 0.D0) GO TO 80
T = 1.D0/T
DO 30 I = KP1,N
30 A(I,K) = -A(I,K)*T
DO 50 J = KP1,N
T = A(M,J)
A(M,J) = A(K,J)
A(K,J) = T
IF (T .EQ. 0.D0) GO TO 50
DO 40 I = KP1,N
40 A(I,J) = A(I,J) + A(I,K)*T
50 CONTINUE
60 CONTINUE
70 K = N
IF (A(N,N) .EQ. 0.D0) GO TO 80
RETURN
80 IER = K
IP(N) = 0
RETURN
C----------------------- END OF SUBROUTINE DEC -------------------------
END
C
C
SUBROUTINE SOL (N, NDIM, A, B, IP)
C VERSION REAL DOUBLE PRECISION
INTEGER N,NDIM,IP,NM1,K,KP1,M,I,KB,KM1
DOUBLE PRECISION A,B,T
DIMENSION A(NDIM,N), B(N), IP(N)
C-----------------------------------------------------------------------
C SOLUTION OF LINEAR SYSTEM, A*X = B .
C INPUT..
C N = ORDER OF MATRIX.
C NDIM = DECLARED DIMENSION OF ARRAY A .
C A = TRIANGULARIZED MATRIX OBTAINED FROM DEC.
C B = RIGHT HAND SIDE VECTOR.
C IP = PIVOT VECTOR OBTAINED FROM DEC.
C DO NOT USE IF DEC HAS SET IER .NE. 0.
C OUTPUT..
C B = SOLUTION VECTOR, X .
C-----------------------------------------------------------------------
IF (N .EQ. 1) GO TO 50
NM1 = N - 1
DO 20 K = 1,NM1
KP1 = K + 1
M = IP(K)
T = B(M)
B(M) = B(K)
B(K) = T
DO 10 I = KP1,N
10 B(I) = B(I) + A(I,K)*T
20 CONTINUE
DO 40 KB = 1,NM1
KM1 = N - KB
K = KM1 + 1
B(K) = B(K)/A(K,K)
T = -B(K)
DO 30 I = 1,KM1
30 B(I) = B(I) + A(I,K)*T
40 CONTINUE
50 B(1) = B(1)/A(1,1)
RETURN
C----------------------- END OF SUBROUTINE SOL -------------------------
END
c
c
SUBROUTINE DECH (N, NDIM, A, LB, IP, IER)
C VERSION REAL DOUBLE PRECISION
INTEGER N,NDIM,IP,IER,NM1,K,KP1,M,I,J,LB,NA
DOUBLE PRECISION A,T
DIMENSION A(NDIM,N), IP(N)
C-----------------------------------------------------------------------
C MATRIX TRIANGULARIZATION BY GAUSSIAN ELIMINATION OF A HESSENBERG
C MATRIX WITH LOWER BANDWIDTH LB
C INPUT..
C N = ORDER OF MATRIX A.
C NDIM = DECLARED DIMENSION OF ARRAY A .
C A = MATRIX TO BE TRIANGULARIZED.
C LB = LOWER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED, LB.GE.1).
C OUTPUT..
C A(I,J), I.LE.J = UPPER TRIANGULAR FACTOR, U .
C A(I,J), I.GT.J = MULTIPLIERS = LOWER TRIANGULAR FACTOR, I - L.
C IP(K), K.LT.N = INDEX OF K-TH PIVOT ROW.
C IP(N) = (-1)**(NUMBER OF INTERCHANGES) OR O .
C IER = 0 IF MATRIX A IS NONSINGULAR, OR K IF FOUND TO BE
C SINGULAR AT STAGE K.
C USE SOLH TO OBTAIN SOLUTION OF LINEAR SYSTEM.
C DETERM(A) = IP(N)*A(1,1)*A(2,2)*...*A(N,N).
C IF IP(N)=O, A IS SINGULAR, SOL WILL DIVIDE BY ZERO.
C
C REFERENCE..
C THIS IS A SLIGHT MODIFICATION OF
C C. B. MOLER, ALGORITHM 423, LINEAR EQUATION SOLVER,
C C.A.C.M. 15 (1972), P. 274.
C-----------------------------------------------------------------------
IER = 0
IP(N) = 1
IF (N .EQ. 1) GO TO 70
NM1 = N - 1
DO 60 K = 1,NM1
KP1 = K + 1
M = K
NA = MIN0(N,LB+K)
DO 10 I = KP1,NA
10 IF (DABS(A(I,K)) .GT. DABS(A(M,K))) M = I
IP(K) = M
T = A(M,K)
IF (M .EQ. K) GO TO 20
IP(N) = -IP(N)
A(M,K) = A(K,K)
A(K,K) = T
20 IF (T .EQ. 0.D0) GO TO 80
T = 1.D0/T
DO 30 I = KP1,NA
30 A(I,K) = -A(I,K)*T
DO 50 J = KP1,N
T = A(M,J)
A(M,J) = A(K,J)
A(K,J) = T
IF (T .EQ. 0.D0) GO TO 50
DO 40 I = KP1,NA
40 A(I,J) = A(I,J) + A(I,K)*T
50 CONTINUE
60 CONTINUE
70 K = N
IF (A(N,N) .EQ. 0.D0) GO TO 80
RETURN
80 IER = K
IP(N) = 0
RETURN
C----------------------- END OF SUBROUTINE DECH ------------------------
END
C
C
SUBROUTINE SOLH (N, NDIM, A, LB, B, IP)
C VERSION REAL DOUBLE PRECISION
INTEGER N,NDIM,IP,NM1,K,KP1,M,I,KB,KM1,LB,NA
DOUBLE PRECISION A,B,T
DIMENSION A(NDIM,N), B(N), IP(N)
C-----------------------------------------------------------------------
C SOLUTION OF LINEAR SYSTEM, A*X = B .
C INPUT..
C N = ORDER OF MATRIX A.
C NDIM = DECLARED DIMENSION OF ARRAY A .
C A = TRIANGULARIZED MATRIX OBTAINED FROM DECH.
C LB = LOWER BANDWIDTH OF A.
C B = RIGHT HAND SIDE VECTOR.
C IP = PIVOT VECTOR OBTAINED FROM DEC.
C DO NOT USE IF DECH HAS SET IER .NE. 0.
C OUTPUT..
C B = SOLUTION VECTOR, X .
C-----------------------------------------------------------------------
IF (N .EQ. 1) GO TO 50
NM1 = N - 1
DO 20 K = 1,NM1
KP1 = K + 1
M = IP(K)
T = B(M)
B(M) = B(K)
B(K) = T
NA = MIN0(N,LB+K)
DO 10 I = KP1,NA
10 B(I) = B(I) + A(I,K)*T
20 CONTINUE
DO 40 KB = 1,NM1
KM1 = N - KB
K = KM1 + 1
B(K) = B(K)/A(K,K)
T = -B(K)
DO 30 I = 1,KM1
30 B(I) = B(I) + A(I,K)*T
40 CONTINUE
50 B(1) = B(1)/A(1,1)
RETURN
C----------------------- END OF SUBROUTINE SOLH ------------------------
END
C
SUBROUTINE DECC (N, NDIM, AR, AI, IP, IER)
C VERSION COMPLEX DOUBLE PRECISION
IMPLICIT REAL*8 (A-H,O-Z)
INTEGER N,NDIM,IP,IER,NM1,K,KP1,M,I,J
DIMENSION AR(NDIM,N), AI(NDIM,N), IP(N)
C-----------------------------------------------------------------------
C MATRIX TRIANGULARIZATION BY GAUSSIAN ELIMINATION
C ------ MODIFICATION FOR COMPLEX MATRICES --------
C INPUT..
C N = ORDER OF MATRIX.
C NDIM = DECLARED DIMENSION OF ARRAYS AR AND AI .
C (AR, AI) = MATRIX TO BE TRIANGULARIZED.
C OUTPUT..
C AR(I,J), I.LE.J = UPPER TRIANGULAR FACTOR, U ; REAL PART.
C AI(I,J), I.LE.J = UPPER TRIANGULAR FACTOR, U ; IMAGINARY PART.
C AR(I,J), I.GT.J = MULTIPLIERS = LOWER TRIANGULAR FACTOR, I - L.
C REAL PART.
C AI(I,J), I.GT.J = MULTIPLIERS = LOWER TRIANGULAR FACTOR, I - L.
C IMAGINARY PART.
C IP(K), K.LT.N = INDEX OF K-TH PIVOT ROW.
C IP(N) = (-1)**(NUMBER OF INTERCHANGES) OR O .
C IER = 0 IF MATRIX A IS NONSINGULAR, OR K IF FOUND TO BE
C SINGULAR AT STAGE K.
C USE SOL TO OBTAIN SOLUTION OF LINEAR SYSTEM.
C IF IP(N)=O, A IS SINGULAR, SOL WILL DIVIDE BY ZERO.
C
C REFERENCE..
C C. B. MOLER, ALGORITHM 423, LINEAR EQUATION SOLVER,
C C.A.C.M. 15 (1972), P. 274.
C-----------------------------------------------------------------------
IER = 0
IP(N) = 1
IF (N .EQ. 1) GO TO 70
NM1 = N - 1
DO 60 K = 1,NM1
KP1 = K + 1
M = K
DO 10 I = KP1,N
10 IF (DABS(AR(I,K))+DABS(AI(I,K)) .GT.
& DABS(AR(M,K))+DABS(AI(M,K))) M = I
IP(K) = M
TR = AR(M,K)
TI = AI(M,K)
IF (M .EQ. K) GO TO 20
IP(N) = -IP(N)
AR(M,K) = AR(K,K)
AI(M,K) = AI(K,K)
AR(K,K) = TR
AI(K,K) = TI
20 IF (DABS(TR)+DABS(TI) .EQ. 0.D0) GO TO 80
DEN=TR*TR+TI*TI
TR=TR/DEN
TI=-TI/DEN
DO 30 I = KP1,N
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
AR(I,K)=-PRODR
AI(I,K)=-PRODI
30 CONTINUE
DO 50 J = KP1,N
TR = AR(M,J)
TI = AI(M,J)
AR(M,J) = AR(K,J)
AI(M,J) = AI(K,J)
AR(K,J) = TR
AI(K,J) = TI
IF (DABS(TR)+DABS(TI) .EQ. 0.D0) GO TO 50
IF (TI .EQ. 0.D0) THEN
DO 40 I = KP1,N
PRODR=AR(I,K)*TR
PRODI=AI(I,K)*TR
AR(I,J) = AR(I,J) + PRODR
AI(I,J) = AI(I,J) + PRODI
40 CONTINUE
GO TO 50
END IF
IF (TR .EQ. 0.D0) THEN
DO 45 I = KP1,N
PRODR=-AI(I,K)*TI
PRODI=AR(I,K)*TI
AR(I,J) = AR(I,J) + PRODR
AI(I,J) = AI(I,J) + PRODI
45 CONTINUE
GO TO 50
END IF
DO 47 I = KP1,N
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
AR(I,J) = AR(I,J) + PRODR
AI(I,J) = AI(I,J) + PRODI
47 CONTINUE
50 CONTINUE
60 CONTINUE
70 K = N
IF (DABS(AR(N,N))+DABS(AI(N,N)) .EQ. 0.D0) GO TO 80
RETURN
80 IER = K
IP(N) = 0
RETURN
C----------------------- END OF SUBROUTINE DECC ------------------------
END
C
C
SUBROUTINE SOLC (N, NDIM, AR, AI, BR, BI, IP)
C VERSION COMPLEX DOUBLE PRECISION
IMPLICIT REAL*8 (A-H,O-Z)
INTEGER N,NDIM,IP,NM1,K,KP1,M,I,KB,KM1
DIMENSION AR(NDIM,N), AI(NDIM,N), BR(N), BI(N), IP(N)
C-----------------------------------------------------------------------
C SOLUTION OF LINEAR SYSTEM, A*X = B .
C INPUT..
C N = ORDER OF MATRIX.
C NDIM = DECLARED DIMENSION OF ARRAYS AR AND AI.
C (AR,AI) = TRIANGULARIZED MATRIX OBTAINED FROM DEC.
C (BR,BI) = RIGHT HAND SIDE VECTOR.
C IP = PIVOT VECTOR OBTAINED FROM DEC.
C DO NOT USE IF DEC HAS SET IER .NE. 0.
C OUTPUT..
C (BR,BI) = SOLUTION VECTOR, X .
C-----------------------------------------------------------------------
IF (N .EQ. 1) GO TO 50
NM1 = N - 1
DO 20 K = 1,NM1
KP1 = K + 1
M = IP(K)
TR = BR(M)
TI = BI(M)
BR(M) = BR(K)
BI(M) = BI(K)
BR(K) = TR
BI(K) = TI
DO 10 I = KP1,N
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
BR(I) = BR(I) + PRODR
BI(I) = BI(I) + PRODI
10 CONTINUE
20 CONTINUE
DO 40 KB = 1,NM1
KM1 = N - KB
K = KM1 + 1
DEN=AR(K,K)*AR(K,K)+AI(K,K)*AI(K,K)
PRODR=BR(K)*AR(K,K)+BI(K)*AI(K,K)
PRODI=BI(K)*AR(K,K)-BR(K)*AI(K,K)
BR(K)=PRODR/DEN
BI(K)=PRODI/DEN
TR = -BR(K)
TI = -BI(K)
DO 30 I = 1,KM1
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
BR(I) = BR(I) + PRODR
BI(I) = BI(I) + PRODI
30 CONTINUE
40 CONTINUE
50 CONTINUE
DEN=AR(1,1)*AR(1,1)+AI(1,1)*AI(1,1)
PRODR=BR(1)*AR(1,1)+BI(1)*AI(1,1)
PRODI=BI(1)*AR(1,1)-BR(1)*AI(1,1)
BR(1)=PRODR/DEN
BI(1)=PRODI/DEN
RETURN
C----------------------- END OF SUBROUTINE SOLC ------------------------
END
C
C
SUBROUTINE DECHC (N, NDIM, AR, AI, LB, IP, IER)
C VERSION COMPLEX DOUBLE PRECISION
IMPLICIT REAL*8 (A-H,O-Z)
INTEGER N,NDIM,IP,IER,NM1,K,KP1,M,I,J
DIMENSION AR(NDIM,N), AI(NDIM,N), IP(N)
C-----------------------------------------------------------------------
C MATRIX TRIANGULARIZATION BY GAUSSIAN ELIMINATION
C ------ MODIFICATION FOR COMPLEX MATRICES --------
C INPUT..
C N = ORDER OF MATRIX.
C NDIM = DECLARED DIMENSION OF ARRAYS AR AND AI .
C (AR, AI) = MATRIX TO BE TRIANGULARIZED.
C OUTPUT..
C AR(I,J), I.LE.J = UPPER TRIANGULAR FACTOR, U ; REAL PART.
C AI(I,J), I.LE.J = UPPER TRIANGULAR FACTOR, U ; IMAGINARY PART.
C AR(I,J), I.GT.J = MULTIPLIERS = LOWER TRIANGULAR FACTOR, I - L.
C REAL PART.
C AI(I,J), I.GT.J = MULTIPLIERS = LOWER TRIANGULAR FACTOR, I - L.
C IMAGINARY PART.
C LB = LOWER BANDWIDTH OF A (DIAGONAL NOT COUNTED), LB.GE.1.
C IP(K), K.LT.N = INDEX OF K-TH PIVOT ROW.
C IP(N) = (-1)**(NUMBER OF INTERCHANGES) OR O .
C IER = 0 IF MATRIX A IS NONSINGULAR, OR K IF FOUND TO BE
C SINGULAR AT STAGE K.
C USE SOL TO OBTAIN SOLUTION OF LINEAR SYSTEM.
C IF IP(N)=O, A IS SINGULAR, SOL WILL DIVIDE BY ZERO.
C
C REFERENCE..
C C. B. MOLER, ALGORITHM 423, LINEAR EQUATION SOLVER,
C C.A.C.M. 15 (1972), P. 274.
C-----------------------------------------------------------------------
IER = 0
IP(N) = 1
IF (LB .EQ. 0) GO TO 70
IF (N .EQ. 1) GO TO 70
NM1 = N - 1
DO 60 K = 1,NM1
KP1 = K + 1
M = K
NA = MIN0(N,LB+K)
DO 10 I = KP1,NA
10 IF (DABS(AR(I,K))+DABS(AI(I,K)) .GT.
& DABS(AR(M,K))+DABS(AI(M,K))) M = I
IP(K) = M
TR = AR(M,K)
TI = AI(M,K)
IF (M .EQ. K) GO TO 20
IP(N) = -IP(N)
AR(M,K) = AR(K,K)
AI(M,K) = AI(K,K)
AR(K,K) = TR
AI(K,K) = TI
20 IF (DABS(TR)+DABS(TI) .EQ. 0.D0) GO TO 80
DEN=TR*TR+TI*TI
TR=TR/DEN
TI=-TI/DEN
DO 30 I = KP1,NA
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
AR(I,K)=-PRODR
AI(I,K)=-PRODI
30 CONTINUE
DO 50 J = KP1,N
TR = AR(M,J)
TI = AI(M,J)
AR(M,J) = AR(K,J)
AI(M,J) = AI(K,J)
AR(K,J) = TR
AI(K,J) = TI
IF (DABS(TR)+DABS(TI) .EQ. 0.D0) GO TO 50
IF (TI .EQ. 0.D0) THEN
DO 40 I = KP1,NA
PRODR=AR(I,K)*TR
PRODI=AI(I,K)*TR
AR(I,J) = AR(I,J) + PRODR
AI(I,J) = AI(I,J) + PRODI
40 CONTINUE
GO TO 50
END IF
IF (TR .EQ. 0.D0) THEN
DO 45 I = KP1,NA
PRODR=-AI(I,K)*TI
PRODI=AR(I,K)*TI
AR(I,J) = AR(I,J) + PRODR
AI(I,J) = AI(I,J) + PRODI
45 CONTINUE
GO TO 50
END IF
DO 47 I = KP1,NA
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
AR(I,J) = AR(I,J) + PRODR
AI(I,J) = AI(I,J) + PRODI
47 CONTINUE
50 CONTINUE
60 CONTINUE
70 K = N
IF (DABS(AR(N,N))+DABS(AI(N,N)) .EQ. 0.D0) GO TO 80
RETURN
80 IER = K
IP(N) = 0
RETURN
C----------------------- END OF SUBROUTINE DECHC -----------------------
END
C
C
SUBROUTINE SOLHC (N, NDIM, AR, AI, LB, BR, BI, IP)
C VERSION COMPLEX DOUBLE PRECISION
IMPLICIT REAL*8 (A-H,O-Z)
INTEGER N,NDIM,IP,NM1,K,KP1,M,I,KB,KM1
DIMENSION AR(NDIM,N), AI(NDIM,N), BR(N), BI(N), IP(N)
C-----------------------------------------------------------------------
C SOLUTION OF LINEAR SYSTEM, A*X = B .
C INPUT..
C N = ORDER OF MATRIX.
C NDIM = DECLARED DIMENSION OF ARRAYS AR AND AI.
C (AR,AI) = TRIANGULARIZED MATRIX OBTAINED FROM DEC.
C (BR,BI) = RIGHT HAND SIDE VECTOR.
C LB = LOWER BANDWIDTH OF A.
C IP = PIVOT VECTOR OBTAINED FROM DEC.
C DO NOT USE IF DEC HAS SET IER .NE. 0.
C OUTPUT..
C (BR,BI) = SOLUTION VECTOR, X .
C-----------------------------------------------------------------------
IF (N .EQ. 1) GO TO 50
NM1 = N - 1
IF (LB .EQ. 0) GO TO 9999
DO 20 K = 1,NM1
KP1 = K + 1
M = IP(K)
TR = BR(M)
TI = BI(M)
BR(M) = BR(K)
BI(M) = BI(K)
BR(K) = TR
BI(K) = TI
DO 10 I = KP1,MIN0(N,LB+K)
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
BR(I) = BR(I) + PRODR
BI(I) = BI(I) + PRODI
10 CONTINUE
20 CONTINUE
9999 CONTINUE
DO 40 KB = 1,NM1
KM1 = N - KB
K = KM1 + 1
DEN=AR(K,K)*AR(K,K)+AI(K,K)*AI(K,K)
PRODR=BR(K)*AR(K,K)+BI(K)*AI(K,K)
PRODI=BI(K)*AR(K,K)-BR(K)*AI(K,K)
BR(K)=PRODR/DEN
BI(K)=PRODI/DEN
TR = -BR(K)
TI = -BI(K)
DO 30 I = 1,KM1
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
BR(I) = BR(I) + PRODR
BI(I) = BI(I) + PRODI
30 CONTINUE
40 CONTINUE
50 CONTINUE
DEN=AR(1,1)*AR(1,1)+AI(1,1)*AI(1,1)
PRODR=BR(1)*AR(1,1)+BI(1)*AI(1,1)
PRODI=BI(1)*AR(1,1)-BR(1)*AI(1,1)
BR(1)=PRODR/DEN
BI(1)=PRODI/DEN
RETURN
C----------------------- END OF SUBROUTINE SOLHC -----------------------
END
C
SUBROUTINE DECB (N, NDIM, A, ML, MU, IP, IER)
REAL*8 A,T
DIMENSION A(NDIM,N), IP(N)
C-----------------------------------------------------------------------
C MATRIX TRIANGULARIZATION BY GAUSSIAN ELIMINATION OF A BANDED
C MATRIX WITH LOWER BANDWIDTH ML AND UPPER BANDWIDTH MU
C INPUT..
C N ORDER OF THE ORIGINAL MATRIX A.
C NDIM DECLARED DIMENSION OF ARRAY A.
C A CONTAINS THE MATRIX IN BAND STORAGE. THE COLUMNS
C OF THE MATRIX ARE STORED IN THE COLUMNS OF A AND
C THE DIAGONALS OF THE MATRIX ARE STORED IN ROWS
C ML+1 THROUGH 2*ML+MU+1 OF A.
C ML LOWER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C MU UPPER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C OUTPUT..
C A AN UPPER TRIANGULAR MATRIX IN BAND STORAGE AND
C THE MULTIPLIERS WHICH WERE USED TO OBTAIN IT.
C IP INDEX VECTOR OF PIVOT INDICES.
C IP(N) (-1)**(NUMBER OF INTERCHANGES) OR O .
C IER = 0 IF MATRIX A IS NONSINGULAR, OR = K IF FOUND TO BE
C SINGULAR AT STAGE K.
C USE SOLB TO OBTAIN SOLUTION OF LINEAR SYSTEM.
C DETERM(A) = IP(N)*A(MD,1)*A(MD,2)*...*A(MD,N) WITH MD=ML+MU+1.
C IF IP(N)=O, A IS SINGULAR, SOLB WILL DIVIDE BY ZERO.
C
C REFERENCE..
C THIS IS A MODIFICATION OF
C C. B. MOLER, ALGORITHM 423, LINEAR EQUATION SOLVER,
C C.A.C.M. 15 (1972), P. 274.
C-----------------------------------------------------------------------
IER = 0
IP(N) = 1
MD = ML + MU + 1
MD1 = MD + 1
JU = 0
IF (ML .EQ. 0) GO TO 70
IF (N .EQ. 1) GO TO 70
IF (N .LT. MU+2) GO TO 7
DO 5 J = MU+2,N
DO 5 I = 1,ML
5 A(I,J) = 0.D0
7 NM1 = N - 1
DO 60 K = 1,NM1
KP1 = K + 1
M = MD
MDL = MIN(ML,N-K) + MD
DO 10 I = MD1,MDL
10 IF (DABS(A(I,K)) .GT. DABS(A(M,K))) M = I
IP(K) = M + K - MD
T = A(M,K)
IF (M .EQ. MD) GO TO 20
IP(N) = -IP(N)
A(M,K) = A(MD,K)
A(MD,K) = T
20 IF (T .EQ. 0.D0) GO TO 80
T = 1.D0/T
DO 30 I = MD1,MDL
30 A(I,K) = -A(I,K)*T
JU = MIN0(MAX0(JU,MU+IP(K)),N)
MM = MD
IF (JU .LT. KP1) GO TO 60
DO 50 J = KP1,JU
M = M - 1
MM = MM - 1
T = A(M,J)
IF (M .EQ. MM) GO TO 35
A(M,J) = A(MM,J)
A(MM,J) = T
35 IF (T .EQ. 0.D0) GO TO 50
JK = J - K
DO 40 I = MD1,MDL
IJK = I - JK
40 A(IJK,J) = A(IJK,J) + A(I,K)*T
50 CONTINUE
60 CONTINUE
70 K = N
IF (A(MD,N) .EQ. 0.D0) GO TO 80
RETURN
80 IER = K
IP(N) = 0
RETURN
C----------------------- END OF SUBROUTINE DECB ------------------------
END
C
C
SUBROUTINE SOLB (N, NDIM, A, ML, MU, B, IP)
REAL*8 A,B,T
DIMENSION A(NDIM,N), B(N), IP(N)
C-----------------------------------------------------------------------
C SOLUTION OF LINEAR SYSTEM, A*X = B .
C INPUT..
C N ORDER OF MATRIX A.
C NDIM DECLARED DIMENSION OF ARRAY A .
C A TRIANGULARIZED MATRIX OBTAINED FROM DECB.
C ML LOWER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C MU UPPER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C B RIGHT HAND SIDE VECTOR.
C IP PIVOT VECTOR OBTAINED FROM DECB.
C DO NOT USE IF DECB HAS SET IER .NE. 0.
C OUTPUT..
C B SOLUTION VECTOR, X .
C-----------------------------------------------------------------------
MD = ML + MU + 1
MD1 = MD + 1
MDM = MD - 1
IF (ML .EQ. 0) GO TO 9998
IF (N .EQ. 1) GO TO 50
NM1 = N - 1
DO 20 K = 1,NM1
M = IP(K)
T = B(M)
B(M) = B(K)
B(K) = T
MDL = MIN(ML,N-K) + MD
DO 10 I = MD1,MDL
IMD = I + K - MD
10 B(IMD) = B(IMD) + A(I,K)*T
20 CONTINUE
9998 CONTINUE
DO 40 KB = 1,NM1
K = N + 1 - KB
B(K) = B(K)/A(MD,K)
T = -B(K)
KMD = MD - K
LM = MAX0(1,KMD+1)
DO 30 I = LM,MDM
IMD = I - KMD
30 B(IMD) = B(IMD) + A(I,K)*T
40 CONTINUE
50 B(1) = B(1)/A(MD,1)
RETURN
C----------------------- END OF SUBROUTINE SOLB ------------------------
END
C
SUBROUTINE DECBC (N, NDIM, AR, AI, ML, MU, IP, IER)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION AR(NDIM,N), AI(NDIM,N), IP(N)
C-----------------------------------------------------------------------
C MATRIX TRIANGULARIZATION BY GAUSSIAN ELIMINATION OF A BANDED COMPLEX
C MATRIX WITH LOWER BANDWIDTH ML AND UPPER BANDWIDTH MU
C INPUT..
C N ORDER OF THE ORIGINAL MATRIX A.
C NDIM DECLARED DIMENSION OF ARRAY A.
C AR, AI CONTAINS THE MATRIX IN BAND STORAGE. THE COLUMNS
C OF THE MATRIX ARE STORED IN THE COLUMNS OF AR (REAL
C PART) AND AI (IMAGINARY PART) AND
C THE DIAGONALS OF THE MATRIX ARE STORED IN ROWS
C ML+1 THROUGH 2*ML+MU+1 OF AR AND AI.
C ML LOWER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C MU UPPER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C OUTPUT..
C AR, AI AN UPPER TRIANGULAR MATRIX IN BAND STORAGE AND
C THE MULTIPLIERS WHICH WERE USED TO OBTAIN IT.
C IP INDEX VECTOR OF PIVOT INDICES.
C IP(N) (-1)**(NUMBER OF INTERCHANGES) OR O .
C IER = 0 IF MATRIX A IS NONSINGULAR, OR = K IF FOUND TO BE
C SINGULAR AT STAGE K.
C USE SOLBC TO OBTAIN SOLUTION OF LINEAR SYSTEM.
C DETERM(A) = IP(N)*A(MD,1)*A(MD,2)*...*A(MD,N) WITH MD=ML+MU+1.
C IF IP(N)=O, A IS SINGULAR, SOLBC WILL DIVIDE BY ZERO.
C
C REFERENCE..
C THIS IS A MODIFICATION OF
C C. B. MOLER, ALGORITHM 423, LINEAR EQUATION SOLVER,
C C.A.C.M. 15 (1972), P. 274.
C-----------------------------------------------------------------------
IER = 0
IP(N) = 1
MD = ML + MU + 1
MD1 = MD + 1
JU = 0
IF (ML .EQ. 0) GO TO 70
IF (N .EQ. 1) GO TO 70
IF (N .LT. MU+2) GO TO 7
DO 5 J = MU+2,N
DO 5 I = 1,ML
AR(I,J) = 0.D0
AI(I,J) = 0.D0
5 CONTINUE
7 NM1 = N - 1
DO 60 K = 1,NM1
KP1 = K + 1
M = MD
MDL = MIN(ML,N-K) + MD
DO 10 I = MD1,MDL
10 IF (DABS(AR(I,K))+DABS(AI(I,K)) .GT.
& DABS(AR(M,K))+DABS(AI(M,K))) M = I
IP(K) = M + K - MD
TR = AR(M,K)
TI = AI(M,K)
IF (M .EQ. MD) GO TO 20
IP(N) = -IP(N)
AR(M,K) = AR(MD,K)
AI(M,K) = AI(MD,K)
AR(MD,K) = TR
AI(MD,K) = TI
20 IF (DABS(TR)+DABS(TI) .EQ. 0.D0) GO TO 80
DEN=TR*TR+TI*TI
TR=TR/DEN
TI=-TI/DEN
DO 30 I = MD1,MDL
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
AR(I,K)=-PRODR
AI(I,K)=-PRODI
30 CONTINUE
JU = MIN0(MAX0(JU,MU+IP(K)),N)
MM = MD
IF (JU .LT. KP1) GO TO 60
DO 50 J = KP1,JU
M = M - 1
MM = MM - 1
TR = AR(M,J)
TI = AI(M,J)
IF (M .EQ. MM) GO TO 35
AR(M,J) = AR(MM,J)
AI(M,J) = AI(MM,J)
AR(MM,J) = TR
AI(MM,J) = TI
35 IF (DABS(TR)+DABS(TI) .EQ. 0.D0) GO TO 50
JK = J - K
IF (TI .EQ. 0.D0) THEN
DO 40 I = MD1,MDL
IJK = I - JK
PRODR=AR(I,K)*TR
PRODI=AI(I,K)*TR
AR(IJK,J) = AR(IJK,J) + PRODR
AI(IJK,J) = AI(IJK,J) + PRODI
40 CONTINUE
GO TO 50
END IF
IF (TR .EQ. 0.D0) THEN
DO 45 I = MD1,MDL
IJK = I - JK
PRODR=-AI(I,K)*TI
PRODI=AR(I,K)*TI
AR(IJK,J) = AR(IJK,J) + PRODR
AI(IJK,J) = AI(IJK,J) + PRODI
45 CONTINUE
GO TO 50
END IF
DO 47 I = MD1,MDL
IJK = I - JK
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
AR(IJK,J) = AR(IJK,J) + PRODR
AI(IJK,J) = AI(IJK,J) + PRODI
47 CONTINUE
50 CONTINUE
60 CONTINUE
70 K = N
IF (DABS(AR(MD,N))+DABS(AI(MD,N)) .EQ. 0.D0) GO TO 80
RETURN
80 IER = K
IP(N) = 0
RETURN
C----------------------- END OF SUBROUTINE DECBC ------------------------
END
C
C
SUBROUTINE SOLBC (N, NDIM, AR, AI, ML, MU, BR, BI, IP)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION AR(NDIM,N), AI(NDIM,N), BR(N), BI(N), IP(N)
C-----------------------------------------------------------------------
C SOLUTION OF LINEAR SYSTEM, A*X = B ,
C VERSION BANDED AND COMPLEX-DOUBLE PRECISION.
C INPUT..
C N ORDER OF MATRIX A.
C NDIM DECLARED DIMENSION OF ARRAY A .
C AR, AI TRIANGULARIZED MATRIX OBTAINED FROM DECB (REAL AND IMAG. PART).
C ML LOWER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C MU UPPER BANDWIDTH OF A (DIAGONAL IS NOT COUNTED).
C BR, BI RIGHT HAND SIDE VECTOR (REAL AND IMAG. PART).
C IP PIVOT VECTOR OBTAINED FROM DECBC.
C DO NOT USE IF DECB HAS SET IER .NE. 0.
C OUTPUT..
C BR, BI SOLUTION VECTOR, X (REAL AND IMAG. PART).
C-----------------------------------------------------------------------
MD = ML + MU + 1
MD1 = MD + 1
MDM = MD - 1
IF (ML .EQ. 0) GO TO 9997
IF (N .EQ. 1) GO TO 50
NM1 = N - 1
DO 20 K = 1,NM1
M = IP(K)
TR = BR(M)
TI = BI(M)
BR(M) = BR(K)
BI(M) = BI(K)
BR(K) = TR
BI(K) = TI
MDL = MIN(ML,N-K) + MD
DO 10 I = MD1,MDL
IMD = I + K - MD
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
BR(IMD) = BR(IMD) + PRODR
BI(IMD) = BI(IMD) + PRODI
10 CONTINUE
20 CONTINUE
9997 CONTINUE
DO 40 KB = 1,NM1
K = N + 1 - KB
DEN=AR(MD,K)*AR(MD,K)+AI(MD,K)*AI(MD,K)
PRODR=BR(K)*AR(MD,K)+BI(K)*AI(MD,K)
PRODI=BI(K)*AR(MD,K)-BR(K)*AI(MD,K)
BR(K)=PRODR/DEN
BI(K)=PRODI/DEN
TR = -BR(K)
TI = -BI(K)
KMD = MD - K
LM = MAX0(1,KMD+1)
DO 30 I = LM,MDM
IMD = I - KMD
PRODR=AR(I,K)*TR-AI(I,K)*TI
PRODI=AI(I,K)*TR+AR(I,K)*TI
BR(IMD) = BR(IMD) + PRODR
BI(IMD) = BI(IMD) + PRODI
30 CONTINUE
40 CONTINUE
DEN=AR(MD,1)*AR(MD,1)+AI(MD,1)*AI(MD,1)
PRODR=BR(1)*AR(MD,1)+BI(1)*AI(MD,1)
PRODI=BI(1)*AR(MD,1)-BR(1)*AI(MD,1)
BR(1)=PRODR/DEN
BI(1)=PRODI/DEN
50 CONTINUE
RETURN
C----------------------- END OF SUBROUTINE SOLBC ------------------------
END
c
C
subroutine elmhes(nm,n,low,igh,a,int)
C
integer i,j,m,n,la,nm,igh,kp1,low,mm1,mp1
real*8 a(nm,n)
real*8 x,y
real*8 dabs
integer int(igh)
C
C this subroutine is a translation of the algol procedure elmhes,
C num. math. 12, 349-368(1968) by martin and wilkinson.
C handbook for auto. comp., vol.ii-linear algebra, 339-358(1971).
C
C given a real general matrix, this subroutine
C reduces a submatrix situated in rows and columns
C low through igh to upper hessenberg form by
C stabilized elementary similarity transformations.
C
C on input:
C
C nm must be set to the row dimension of two-dimensional
C array parameters as declared in the calling program
C dimension statement;
C
C n is the order of the matrix;
C
C low and igh are integers determined by the balancing
C subroutine balanc. if balanc has not been used,
C set low=1, igh=n;
C
C a contains the input matrix.
C
C on output:
C
C a contains the hessenberg matrix. the multipliers
C which were used in the reduction are stored in the
C remaining triangle under the hessenberg matrix;
C
C int contains information on the rows and columns
C interchanged in the reduction.
C only elements low through igh are used.
C
C questions and comments should be directed to b. s. garbow,
C applied mathematics division, argonne national laboratory
C
C ------------------------------------------------------------------
C
la = igh - 1
kp1 = low + 1
if (la .lt. kp1) go to 200
C
do 180 m = kp1, la
mm1 = m - 1
x = 0.0d0
i = m
C
do 100 j = m, igh
if (dabs(a(j,mm1)) .le. dabs(x)) go to 100
x = a(j,mm1)
i = j
100 continue
C
int(m) = i
if (i .eq. m) go to 130
C :::::::::: interchange rows and columns of a ::::::::::
do 110 j = mm1, n
y = a(i,j)
a(i,j) = a(m,j)
a(m,j) = y
110 continue
C
do 120 j = 1, igh
y = a(j,i)
a(j,i) = a(j,m)
a(j,m) = y
120 continue
C :::::::::: end interchange ::::::::::
130 if (x .eq. 0.0d0) go to 180
mp1 = m + 1
C
do 160 i = mp1, igh
y = a(i,mm1)
if (y .eq. 0.0d0) go to 160
y = y / x
a(i,mm1) = y
C
do 140 j = m, n
140 a(i,j) = a(i,j) - y * a(m,j)
C
do 150 j = 1, igh
150 a(j,m) = a(j,m) + y * a(j,i)
C
160 continue
C
180 continue
C
200 return
C :::::::::: last card of elmhes ::::::::::
end
Simulation Platform