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tools_math.f90
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! Copyright (c) 2013-2016 Alberto Otero de la Roza
! <aoterodelaroza@gmail.com>, Felix Kannemann
! <felix.kannemann@dal.ca>, Erin R. Johnson <erin.johnson@dal.ca>,
! Ross M. Dickson <ross.dickson@dal.ca>, Hartmut Schmider
! <hs7@post.queensu.ca>, and Axel D. Becke <axel.becke@dal.ca>
!
! postg is free software: you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation, either version 3 of the License, or
! (at your option) any later version.
!
! This program is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with this program. If not, see <http://www.gnu.org/licenses/>.
module tools_math
! spline and linpack
public :: spline
! lebedev for postg
public :: LD0006, LD0014, LD0026, LD0038, LD0050, LD0074, LD0086,&
LD0110, LD0146, LD0170, LD0194, LD0230, LD0266, LD0302,&
LD0350, LD0434, LD0590, LD0770, LD0974, LD1202, LD1454,&
LD1730, LD2030, LD2354, LD2702, LD3074, LD3470, LD3890,&
LD4334, LD4802, LD5294, LD5810
contains
SUBROUTINE SPLINE (H,Y,A,B,C,N,BCR)
!
! NATURAL CUBIC SPLINE THROUGH DATA POINTS ON THE UNIFORM R-MESH.
! RETURNS COEFFICIENTS A,B,C DEFINING THE CUBIC POLYNOMIAL TO THE
! RIGHT OF EACH KNOT (INCLUDING A(0),B(0),C(0)).
!
! INPUT:
! H - INTERVAL BETWEEN KNOTS
! Y - ARRAY OF ORDINATE VALUES
! N - THE NUMBER OF DATA POINTS
! BCR - VALUE OF FUNCTION AT R=INFINITY
!
! OUTPUT:
! A - ARRAY OF COEFFICIENTS OF X
! B - ARRAY OF COEFFICIENTS OF X**2
! C - ARRAY OF COEFFICIENTS OF X**3
!
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION Y(0:N),A(0:N),B(0:N),C(0:N)
DIMENSION D(N),E(N),BB(N)
intent(in) :: H, Y, N, BCR
intent(out) :: A, B, C
THRD=1.D0/3.D0
THRD2=2.D0*THRD
HINV=1.D0/H
H2INV=HINV*HINV
DO I=1,N-1
D(I)=4.D0
E(I)=1.D0
BB(I)=3.D0*H2INV*(Y(I-1)-2.D0*Y(I)+Y(I+1))
enddo
D(N)=4.D0
E(N)=0.D0
BB(N)=3.D0*H2INV*(Y(N-1)-2.D0*Y(N)+BCR)
CALL DPTSL(N,D,E,BB)
A(0)=HINV*(Y(1)-Y(0))-THRD*H*BB(1)
B(0)=0.D0
C(0)=THRD*HINV*BB(1)
DO I=1,N-1
A(I)=HINV*(Y(I+1)-Y(I))-THRD2*H*BB(I)-THRD*H*BB(I+1)
B(I)=BB(I)
C(I)=THRD*HINV*(BB(I+1)-BB(I))
enddo
A(N)=HINV*(BCR-Y(N))-THRD2*H*BB(N)
B(N)=BB(N)
C(N)=-THRD*HINV*BB(N)
END SUBROUTINE SPLINE
SUBROUTINE DGECO(A,LDA,N,IPVT,RCOND,Z)
INTEGER LDA,N,IPVT(1)
REAL*8 A(LDA,1),Z(1)
REAL*8 RCOND
!
! DGECO FACTORS A REAL*8 MATRIX BY GAUSSIAN ELIMINATION
! AND ESTIMATES THE CONDITION OF THE MATRIX.
!
! IF RCOND IS NOT NEEDED, DGEFA IS SLIGHTLY FASTER.
! TO SOLVE A*X = B , FOLLOW DGECO BY DGESL.
! TO COMPUTE INVERSE(A)*C , FOLLOW DGECO BY DGESL.
! TO COMPUTE DETERMINANT(A) , FOLLOW DGECO BY DGEDI.
! TO COMPUTE INVERSE(A) , FOLLOW DGECO BY DGEDI.
!
! ON ENTRY
!
! A REAL*8(LDA, N)
! THE MATRIX TO BE FACTORED.
!
! LDA INTEGER
! THE LEADING DIMENSION OF THE ARRAY A .
!
! N INTEGER
! THE ORDER OF THE MATRIX A .
!
! ON RETURN
!
! A AN UPPER TRIANGULAR MATRIX AND THE MULTIPLIERS
! WHICH WERE USED TO OBTAIN IT.
! THE FACTORIZATION CAN BE WRITTEN A = L*U WHERE
! L IS A PRODUCT OF PERMUTATION AND UNIT LOWER
! TRIANGULAR MATRICES AND U IS UPPER TRIANGULAR.
!
! IPVT INTEGER(N)
! AN INTEGER VECTOR OF PIVOT INDICES.
!
! RCOND REAL*8
! AN ESTIMATE OF THE RECIPROCAL CONDITION OF A .
! FOR THE SYSTEM A*X = B , RELATIVE PERTURBATIONS
! IN A AND B OF SIZE EPSILON MAY CAUSE
! RELATIVE PERTURBATIONS IN X OF SIZE EPSILON/RCOND .
! IF RCOND IS SO SMALL THAT THE LOGICAL EXPRESSION
! 1.0 + RCOND .EQ. 1.0
! IS TRUE, THEN A MAY BE SINGULAR TO WORKING
! PRECISION. IN PARTICULAR, RCOND IS ZERO IF
! EXACT SINGULARITY IS DETECTED OR THE ESTIMATE
! UNDERFLOWS.
!
! Z REAL*8(N)
! A WORK VECTOR WHOSE CONTENTS ARE USUALLY UNIMPORTANT.
! IF A IS CLOSE TO A SINGULAR MATRIX, THEN Z IS
! AN APPROXIMATE NULL VECTOR IN THE SENSE THAT
! NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
!
! LINPACK. THIS VERSION DATED 08/14/78 .
! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB.
!
! SUBROUTINES AND FUNCTIONS
!
! LINPACK DGEFA
! BLAS DAXPY,DDOT,DSCAL,DASUM
! FORTRAN DABS,DMAX1,DSIGN
!
! INTERNAL VARIABLES
!
REAL*8 EK,T,WK,WKM
REAL*8 ANORM,S,SM,YNORM
INTEGER INFO,J,K,KB,KP1,L
!
!
! COMPUTE 1-NORM OF A
!
ANORM = 0.0D0
DO 10 J = 1, N
ANORM = DMAX1(ANORM,DASUM(N,A(1,J),1))
10 CONTINUE
!
! FACTOR
!
CALL DGEFA(A,LDA,N,IPVT,INFO)
!
! RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) .
! ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND TRANS(A)*Y = E .
! TRANS(A) IS THE TRANSPOSE OF A . THE COMPONENTS OF E ARE
! CHOSEN TO CAUSE MAXIMUM LOCAL GROWTH IN THE ELEMENTS OF W WHERE
! TRANS(U)*W = E . THE VECTORS ARE FREQUENTLY RESCALED TO AVOID
! OVERFLOW.
!
! SOLVE TRANS(U)*W = E
!
EK = 1.0D0
DO 20 J = 1, N
Z(J) = 0.0D0
20 CONTINUE
DO 100 K = 1, N
IF (Z(K) .NE. 0.0D0) EK = DSIGN(EK,-Z(K))
IF (DABS(EK-Z(K)) .LE. DABS(A(K,K))) GO TO 30
S = DABS(A(K,K))/DABS(EK-Z(K))
CALL DSCAL(N,S,Z,1)
EK = S*EK
30 CONTINUE
WK = EK - Z(K)
WKM = -EK - Z(K)
S = DABS(WK)
SM = DABS(WKM)
IF (A(K,K) .EQ. 0.0D0) GO TO 40
WK = WK/A(K,K)
WKM = WKM/A(K,K)
GO TO 50
40 CONTINUE
WK = 1.0D0
WKM = 1.0D0
50 CONTINUE
KP1 = K + 1
IF (KP1 .GT. N) GO TO 90
DO 60 J = KP1, N
SM = SM + DABS(Z(J)+WKM*A(K,J))
Z(J) = Z(J) + WK*A(K,J)
S = S + DABS(Z(J))
60 CONTINUE
IF (S .GE. SM) GO TO 80
T = WKM - WK
WK = WKM
DO 70 J = KP1, N
Z(J) = Z(J) + T*A(K,J)
70 CONTINUE
80 CONTINUE
90 CONTINUE
Z(K) = WK
100 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
!
! SOLVE TRANS(L)*Y = W
!
DO 120 KB = 1, N
K = N + 1 - KB
IF (K .LT. N) Z(K) = Z(K) + DDOT(N-K,A(K+1,K),1,Z(K+1),1)
IF (DABS(Z(K)) .LE. 1.0D0) GO TO 110
S = 1.0D0/DABS(Z(K))
CALL DSCAL(N,S,Z,1)
110 CONTINUE
L = IPVT(K)
T = Z(L)
Z(L) = Z(K)
Z(K) = T
120 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
!
YNORM = 1.0D0
!
! SOLVE L*V = Y
!
DO 140 K = 1, N
L = IPVT(K)
T = Z(L)
Z(L) = Z(K)
Z(K) = T
IF (K .LT. N) CALL DAXPY(N-K,T,A(K+1,K),1,Z(K+1),1)
IF (DABS(Z(K)) .LE. 1.0D0) GO TO 130
S = 1.0D0/DABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
130 CONTINUE
140 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
!
! SOLVE U*Z = V
!
DO 160 KB = 1, N
K = N + 1 - KB
IF (DABS(Z(K)) .LE. DABS(A(K,K))) GO TO 150
S = DABS(A(K,K))/DABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
150 CONTINUE
IF (A(K,K) .NE. 0.0D0) Z(K) = Z(K)/A(K,K)
IF (A(K,K) .EQ. 0.0D0) Z(K) = 1.0D0
T = -Z(K)
CALL DAXPY(K-1,T,A(1,K),1,Z(1),1)
160 CONTINUE
! MAKE ZNORM = 1.0
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
!
IF (ANORM .NE. 0.0D0) RCOND = YNORM/ANORM
IF (ANORM .EQ. 0.0D0) RCOND = 0.0D0
RETURN
END subroutine
SUBROUTINE DGEFA(A,LDA,N,IPVT,INFO)
INTEGER LDA,N,IPVT(1),INFO
REAL*8 A(LDA,1)
!
! DGEFA FACTORS A REAL*8 MATRIX BY GAUSSIAN ELIMINATION.
!
! DGEFA IS USUALLY CALLED BY DGECO, BUT IT CAN BE CALLED
! DIRECTLY WITH A SAVING IN TIME IF RCOND IS NOT NEEDED.
! (TIME FOR DGECO) = (1 + 9/N)*(TIME FOR DGEFA) .
!
! ON ENTRY
!
! A REAL*8(LDA, N)
! THE MATRIX TO BE FACTORED.
!
! LDA INTEGER
! THE LEADING DIMENSION OF THE ARRAY A .
!
! N INTEGER
! THE ORDER OF THE MATRIX A .
!
! ON RETURN
!
! A AN UPPER TRIANGULAR MATRIX AND THE MULTIPLIERS
! WHICH WERE USED TO OBTAIN IT.
! THE FACTORIZATION CAN BE WRITTEN A = L*U WHERE
! L IS A PRODUCT OF PERMUTATION AND UNIT LOWER
! TRIANGULAR MATRICES AND U IS UPPER TRIANGULAR.
!
! IPVT INTEGER(N)
! AN INTEGER VECTOR OF PIVOT INDICES.
!
! INFO INTEGER
! = 0 NORMAL VALUE.
! = K IF U(K,K) .EQ. 0.0 . THIS IS NOT AN ERROR
! CONDITION FOR THIS SUBROUTINE, BUT IT DOES
! INDICATE THAT DGESL OR DGEDI WILL DIVIDE BY ZERO
! IF CALLED. USE RCOND IN DGECO FOR A RELIABLE
! INDICATION OF SINGULARITY.
!
! LINPACK. THIS VERSION DATED 08/14/78 .
! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB.
!
! SUBROUTINES AND FUNCTIONS
!
! BLAS DAXPY,DSCAL,IDAMAX
!
! INTERNAL VARIABLES
!
REAL*8 T
INTEGER J,K,KP1,L,NM1
!
!
! GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING
!
INFO = 0
NM1 = N - 1
IF (NM1 .LT. 1) GO TO 70
DO 60 K = 1, NM1
KP1 = K + 1
!
! FIND L = PIVOT INDEX
!
L = IDAMAX(N-K+1,A(K,K),1) + K - 1
IPVT(K) = L
!
! ZERO PIVOT IMPLIES THIS COLUMN ALREADY TRIANGULARIZED
!
IF (A(L,K) .EQ. 0.0D0) GO TO 40
!
! INTERCHANGE IF NECESSARY
!
IF (L .EQ. K) GO TO 10
T = A(L,K)
A(L,K) = A(K,K)
A(K,K) = T
10 CONTINUE
!
! COMPUTE MULTIPLIERS
!
T = -1.0D0/A(K,K)
CALL DSCAL(N-K,T,A(K+1,K),1)
!
! ROW ELIMINATION WITH COLUMN INDEXING
!
DO 30 J = KP1, N
T = A(L,J)
IF (L .EQ. K) GO TO 20
A(L,J) = A(K,J)
A(K,J) = T
20 CONTINUE
CALL DAXPY(N-K,T,A(K+1,K),1,A(K+1,J),1)
30 CONTINUE
GO TO 50
40 CONTINUE
INFO = K
50 CONTINUE
60 CONTINUE
70 CONTINUE
IPVT(N) = N
IF (A(N,N) .EQ. 0.0D0) INFO = N
RETURN
END subroutine
SUBROUTINE DGESL(A,LDA,N,IPVT,B,JOB)
INTEGER LDA,N,IPVT(1),JOB
REAL*8 A(LDA,1),B(1)
!
! DGESL SOLVES THE REAL*8 SYSTEM
! A * X = B OR TRANS(A) * X = B
! USING THE FACTORS COMPUTED BY DGECO OR DGEFA.
!
! ON ENTRY
!
! A REAL*8(LDA, N)
! THE OUTPUT FROM DGECO OR DGEFA.
!
! LDA INTEGER
! THE LEADING DIMENSION OF THE ARRAY A .
!
! N INTEGER
! THE ORDER OF THE MATRIX A .
!
! IPVT INTEGER(N)
! THE PIVOT VECTOR FROM DGECO OR DGEFA.
!
! B REAL*8(N)
! THE RIGHT HAND SIDE VECTOR.
!
! JOB INTEGER
! = 0 TO SOLVE A*X = B ,
! = NONZERO TO SOLVE TRANS(A)*X = B WHERE
! TRANS(A) IS THE TRANSPOSE.
!
! ON RETURN
!
! B THE SOLUTION VECTOR X .
!
! ERROR CONDITION
!
! A DIVISION BY ZERO WILL OCCUR IF THE INPUT FACTOR CONTAINS A
! ZERO ON THE DIAGONAL. TECHNICALLY THIS INDICATES SINGULARITY
! BUT IT IS OFTEN CAUSED BY IMPROPER ARGUMENTS OR IMPROPER
! SETTING OF LDA . IT WILL NOT OCCUR IF THE SUBROUTINES ARE
! CALLED CORRECTLY AND IF DGECO HAS SET RCOND .GT. 0.0
! OR DGEFA HAS SET INFO .EQ. 0 .
!
! TO COMPUTE INVERSE(A) * C WHERE C IS A MATRIX
! WITH P COLUMNS
! CALL DGECO(A,LDA,N,IPVT,RCOND,Z)
! IF (RCOND IS TOO SMALL) GO TO ...
! DO 10 J = 1, P
! CALL DGESL(A,LDA,N,IPVT,C(1,J),0)
! 10 CONTINUE
!
! LINPACK. THIS VERSION DATED 08/14/78 .
! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB.
!
! SUBROUTINES AND FUNCTIONS
!
! BLAS DAXPY,DDOT
!
! INTERNAL VARIABLES
!
REAL*8 T
INTEGER K,KB,L,NM1
!
NM1 = N - 1
IF (JOB .NE. 0) GO TO 50
!
! JOB = 0 , SOLVE A * X = B
! FIRST SOLVE L*Y = B
!
IF (NM1 .LT. 1) GO TO 30
DO 20 K = 1, NM1
L = IPVT(K)
T = B(L)
IF (L .EQ. K) GO TO 10
B(L) = B(K)
B(K) = T
10 CONTINUE
CALL DAXPY(N-K,T,A(K+1,K),1,B(K+1),1)
20 CONTINUE
30 CONTINUE
!
! NOW SOLVE U*X = Y
!
DO 40 KB = 1, N
K = N + 1 - KB
B(K) = B(K)/A(K,K)
T = -B(K)
CALL DAXPY(K-1,T,A(1,K),1,B(1),1)
40 CONTINUE
GO TO 100
50 CONTINUE
!
! JOB = NONZERO, SOLVE TRANS(A) * X = B
! FIRST SOLVE TRANS(U)*Y = B
!
DO 60 K = 1, N
T = DDOT(K-1,A(1,K),1,B(1),1)
B(K) = (B(K) - T)/A(K,K)
60 CONTINUE
!
! NOW SOLVE TRANS(L)*X = Y
!
IF (NM1 .LT. 1) GO TO 90
DO 80 KB = 1, NM1
K = N - KB
B(K) = B(K) + DDOT(N-K,A(K+1,K),1,B(K+1),1)
L = IPVT(K)
IF (L .EQ. K) GO TO 70
T = B(L)
B(L) = B(K)
B(K) = T
70 CONTINUE
80 CONTINUE
90 CONTINUE
100 CONTINUE
RETURN
END subroutine
SUBROUTINE DGEDI(A,LDA,N,IPVT,DET,WORK,JOB)
INTEGER LDA,N,IPVT(1),JOB
REAL*8 A(LDA,1),DET(2),WORK(1)
!
! DGEDI COMPUTES THE DETERMINANT AND INVERSE OF A MATRIX
! USING THE FACTORS COMPUTED BY DGECO OR DGEFA.
!
! ON ENTRY
!
! A REAL*8(LDA, N)
! THE OUTPUT FROM DGECO OR DGEFA.
!
! LDA INTEGER
! THE LEADING DIMENSION OF THE ARRAY A .
!
! N INTEGER
! THE ORDER OF THE MATRIX A .
!
! IPVT INTEGER(N)
! THE PIVOT VECTOR FROM DGECO OR DGEFA.
!
! WORK REAL*8(N)
! WORK VECTOR. CONTENTS DESTROYED.
!
! JOB INTEGER
! = 11 BOTH DETERMINANT AND INVERSE.
! = 01 INVERSE ONLY.
! = 10 DETERMINANT ONLY.
!
! ON RETURN
!
! A INVERSE OF ORIGINAL MATRIX IF REQUESTED.
! OTHERWISE UNCHANGED.
!
! DET REAL*8(2)
! DETERMINANT OF ORIGINAL MATRIX IF REQUESTED.
! OTHERWISE NOT REFERENCED.
! DETERMINANT = DET(1) * 10.0**DET(2)
! WITH 1.0 .LE. DABS(DET(1)) .LT. 10.0
! OR DET(1) .EQ. 0.0 .
!
! ERROR CONDITION
!
! A DIVISION BY ZERO WILL OCCUR IF THE INPUT FACTOR CONTAINS
! A ZERO ON THE DIAGONAL AND THE INVERSE IS REQUESTED.
! IT WILL NOT OCCUR IF THE SUBROUTINES ARE CALLED CORRECTLY
! AND IF DGECO HAS SET RCOND .GT. 0.0 OR DGEFA HAS SET
! INFO .EQ. 0 .
!
! LINPACK. THIS VERSION DATED 08/14/78 .
! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB.
!
! SUBROUTINES AND FUNCTIONS
!
! BLAS DAXPY,DSCAL,DSWAP
! FORTRAN DABS,MOD
!
! INTERNAL VARIABLES
!
REAL*8 T
REAL*8 TEN
INTEGER I,J,K,KB,KP1,L,NM1
!
!
! COMPUTE DETERMINANT
!
IF (JOB/10 .EQ. 0) GO TO 70
DET(1) = 1.0D0
DET(2) = 0.0D0
TEN = 10.0D0
DO 50 I = 1, N
IF (IPVT(I) .NE. I) DET(1) = -DET(1)
DET(1) = A(I,I)*DET(1)
! ...EXIT
IF (DET(1) .EQ. 0.0D0) GO TO 60
10 IF (DABS(DET(1)) .GE. 1.0D0) GO TO 20
DET(1) = TEN*DET(1)
DET(2) = DET(2) - 1.0D0
GO TO 10
20 CONTINUE
30 IF (DABS(DET(1)) .LT. TEN) GO TO 40
DET(1) = DET(1)/TEN
DET(2) = DET(2) + 1.0D0
GO TO 30
40 CONTINUE
50 CONTINUE
60 CONTINUE
70 CONTINUE
!
! COMPUTE INVERSE(U)
!
IF (MOD(JOB,10) .EQ. 0) GO TO 150
DO 100 K = 1, N
A(K,K) = 1.0D0/A(K,K)
T = -A(K,K)
CALL DSCAL(K-1,T,A(1,K),1)
KP1 = K + 1
IF (N .LT. KP1) GO TO 90
DO 80 J = KP1, N
T = A(K,J)
A(K,J) = 0.0D0
CALL DAXPY(K,T,A(1,K),1,A(1,J),1)
80 CONTINUE
90 CONTINUE
100 CONTINUE
!
! FORM INVERSE(U)*INVERSE(L)
!
NM1 = N - 1
IF (NM1 .LT. 1) GO TO 140
DO 130 KB = 1, NM1
K = N - KB
KP1 = K + 1
DO 110 I = KP1, N
WORK(I) = A(I,K)
A(I,K) = 0.0D0
110 CONTINUE
DO 120 J = KP1, N
T = WORK(J)
CALL DAXPY(N,T,A(1,J),1,A(1,K),1)
120 CONTINUE
L = IPVT(K)
IF (L .NE. K) CALL DSWAP(N,A(1,K),1,A(1,L),1)
130 CONTINUE
140 CONTINUE
150 CONTINUE
RETURN
END subroutine
SUBROUTINE DGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)
INTEGER LDA,N,ML,MU,IPVT(1)
REAL*8 ABD(LDA,1),Z(1)
REAL*8 RCOND
!
! DGBCO FACTORS A REAL*8 BAND MATRIX BY GAUSSIAN
! ELIMINATION AND ESTIMATES THE CONDITION OF THE MATRIX.
!
! IF RCOND IS NOT NEEDED, DGBFA IS SLIGHTLY FASTER.
! TO SOLVE A*X = B , FOLLOW DGBCO BY DGBSL.
! TO COMPUTE INVERSE(A)*C , FOLLOW DGBCO BY DGBSL.
! TO COMPUTE DETERMINANT(A) , FOLLOW DGBCO BY DGBDI.
!
! ON ENTRY
!
! ABD REAL*8(LDA, N)
! CONTAINS THE MATRIX IN BAND STORAGE. THE COLUMNS
! OF THE MATRIX ARE STORED IN THE COLUMNS OF ABD AND
! THE DIAGONALS OF THE MATRIX ARE STORED IN ROWS
! ML+1 THROUGH 2*ML+MU+1 OF ABD .
! SEE THE COMMENTS BELOW FOR DETAILS.
!
! LDA INTEGER
! THE LEADING DIMENSION OF THE ARRAY ABD .
! LDA MUST BE .GE. 2*ML + MU + 1 .
!
! N INTEGER
! THE ORDER OF THE ORIGINAL MATRIX.
!
! ML INTEGER
! NUMBER OF DIAGONALS BELOW THE MAIN DIAGONAL.
! 0 .LE. ML .LT. N .
!
! MU INTEGER
! NUMBER OF DIAGONALS ABOVE THE MAIN DIAGONAL.
! 0 .LE. MU .LT. N .
! MORE EFFICIENT IF ML .LE. MU .
!
! ON RETURN
!
! ABD AN UPPER TRIANGULAR MATRIX IN BAND STORAGE AND
! THE MULTIPLIERS WHICH WERE USED TO OBTAIN IT.
! THE FACTORIZATION CAN BE WRITTEN A = L*U WHERE
! L IS A PRODUCT OF PERMUTATION AND UNIT LOWER
! TRIANGULAR MATRICES AND U IS UPPER TRIANGULAR.
!
! IPVT INTEGER(N)
! AN INTEGER VECTOR OF PIVOT INDICES.
!
! RCOND REAL*8
! AN ESTIMATE OF THE RECIPROCAL CONDITION OF A .
! FOR THE SYSTEM A*X = B , RELATIVE PERTURBATIONS
! IN A AND B OF SIZE EPSILON MAY CAUSE
! RELATIVE PERTURBATIONS IN X OF SIZE EPSILON/RCOND .
! IF RCOND IS SO SMALL THAT THE LOGICAL EXPRESSION
! 1.0 + RCOND .EQ. 1.0
! IS TRUE, THEN A MAY BE SINGULAR TO WORKING
! PRECISION. IN PARTICULAR, RCOND IS ZERO IF
! EXACT SINGULARITY IS DETECTED OR THE ESTIMATE
! UNDERFLOWS.
!
! Z REAL*8(N)
! A WORK VECTOR WHOSE CONTENTS ARE USUALLY UNIMPORTANT.
! IF A IS CLOSE TO A SINGULAR MATRIX, THEN Z IS
! AN APPROXIMATE NULL VECTOR IN THE SENSE THAT
! NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
!
! BAND STORAGE
!
! IF A IS A BAND MATRIX, THE FOLLOWING PROGRAM SEGMENT
! WILL SET UP THE INPUT.
!
! ML = (BAND WIDTH BELOW THE DIAGONAL)
! MU = (BAND WIDTH ABOVE THE DIAGONAL)
! M = ML + MU + 1
! DO 20 J = 1, N
! I1 = MAX0(1, J-MU)
! I2 = MIN0(N, J+ML)
! DO 10 I = I1, I2
! K = I - J + M
! ABD(K,J) = A(I,J)
! 10 CONTINUE
! 20 CONTINUE
!
! THIS USES ROWS ML+1 THROUGH 2*ML+MU+1 OF ABD .
! IN ADDITION, THE FIRST ML ROWS IN ABD ARE USED FOR
! ELEMENTS GENERATED DURING THE TRIANGULARIZATION.
! THE TOTAL NUMBER OF ROWS NEEDED IN ABD IS 2*ML+MU+1 .
! THE ML+MU BY ML+MU UPPER LEFT TRIANGLE AND THE
! ML BY ML LOWER RIGHT TRIANGLE ARE NOT REFERENCED.
!
! EXAMPLE.. IF THE ORIGINAL MATRIX IS
!
! 11 12 13 0 0 0
! 21 22 23 24 0 0
! 0 32 33 34 35 0
! 0 0 43 44 45 46
! 0 0 0 54 55 56
! 0 0 0 0 65 66
!
! THEN N = 6, ML = 1, MU = 2, LDA .GE. 5 AND ABD SHOULD CONTAIN
!
! * * * + + + , * = NOT USED
! * * 13 24 35 46 , + = USED FOR PIVOTING
! * 12 23 34 45 56
! 11 22 33 44 55 66
! 21 32 43 54 65 *
!
! LINPACK. THIS VERSION DATED 08/14/78 .
! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB.
!
! SUBROUTINES AND FUNCTIONS
!
! LINPACK DGBFA
! BLAS DAXPY,DDOT,DSCAL,DASUM
! FORTRAN DABS,DMAX1,MAX0,MIN0,DSIGN
!
! INTERNAL VARIABLES
!
REAL*8 EK,T,WK,WKM
REAL*8 ANORM,S,SM,YNORM
INTEGER IS,INFO,J,JU,K,KB,KP1,L,LA,LM,LZ,M,MM
!
!
! COMPUTE 1-NORM OF A
!
ANORM = 0.0D0
L = ML + 1
IS = L + MU
DO 10 J = 1, N
ANORM = DMAX1(ANORM,DASUM(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
!
! FACTOR
!
CALL DGBFA(ABD,LDA,N,ML,MU,IPVT,INFO)
!
! RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) .
! ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND TRANS(A)*Y = E .
! TRANS(A) IS THE TRANSPOSE OF A . THE COMPONENTS OF E ARE
! CHOSEN TO CAUSE MAXIMUM LOCAL GROWTH IN THE ELEMENTS OF W WHERE
! TRANS(U)*W = E . THE VECTORS ARE FREQUENTLY RESCALED TO AVOID
! OVERFLOW.
!
! SOLVE TRANS(U)*W = E
!
EK = 1.0D0
DO 20 J = 1, N
Z(J) = 0.0D0
20 CONTINUE
M = ML + MU + 1
JU = 0
DO 100 K = 1, N
IF (Z(K) .NE. 0.0D0) EK = DSIGN(EK,-Z(K))
IF (DABS(EK-Z(K)) .LE. DABS(ABD(M,K))) GO TO 30
S = DABS(ABD(M,K))/DABS(EK-Z(K))
CALL DSCAL(N,S,Z,1)
EK = S*EK
30 CONTINUE
WK = EK - Z(K)
WKM = -EK - Z(K)
S = DABS(WK)
SM = DABS(WKM)
IF (ABD(M,K) .EQ. 0.0D0) GO TO 40
WK = WK/ABD(M,K)
WKM = WKM/ABD(M,K)
GO TO 50
40 CONTINUE
WK = 1.0D0
WKM = 1.0D0
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 + DABS(Z(J)+WKM*ABD(MM,J))
Z(J) = Z(J) + WK*ABD(MM,J)
S = S + DABS(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 = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
!
! SOLVE TRANS(L)*Y = W
!
DO 120 KB = 1, N
K = N + 1 - KB
LM = MIN0(ML,N-K)
IF (K .LT. N) Z(K) = Z(K) + DDOT(LM,ABD(M+1,K),1,Z(K+1),1)
IF (DABS(Z(K)) .LE. 1.0D0) GO TO 110
S = 1.0D0/DABS(Z(K))
CALL DSCAL(N,S,Z,1)
110 CONTINUE
L = IPVT(K)
T = Z(L)
Z(L) = Z(K)
Z(K) = T
120 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
!
YNORM = 1.0D0
!
! SOLVE L*V = Y
!
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 DAXPY(LM,T,ABD(M+1,K),1,Z(K+1),1)
IF (DABS(Z(K)) .LE. 1.0D0) GO TO 130
S = 1.0D0/DABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
130 CONTINUE
140 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
!
! SOLVE U*Z = W
!
DO 160 KB = 1, N
K = N + 1 - KB
IF (DABS(Z(K)) .LE. DABS(ABD(M,K))) GO TO 150
S = DABS(ABD(M,K))/DABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
150 CONTINUE
IF (ABD(M,K) .NE. 0.0D0) Z(K) = Z(K)/ABD(M,K)
IF (ABD(M,K) .EQ. 0.0D0) Z(K) = 1.0D0
LM = MIN0(K,M) - 1
LA = M - LM
LZ = K - LM
T = -Z(K)
CALL DAXPY(LM,T,ABD(LA,K),1,Z(LZ),1)
160 CONTINUE
! MAKE ZNORM = 1.0
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
!
IF (ANORM .NE. 0.0D0) RCOND = YNORM/ANORM
IF (ANORM .EQ. 0.0D0) RCOND = 0.0D0
RETURN
END subroutine
SUBROUTINE DGBFA(ABD,LDA,N,ML,MU,IPVT,INFO)
INTEGER LDA,N,ML,MU,IPVT(1),INFO
REAL*8 ABD(LDA,1)
!
! DGBFA FACTORS A REAL*8 BAND MATRIX BY ELIMINATION.
!
! DGBFA IS USUALLY CALLED BY DGBCO, BUT IT CAN BE CALLED
! DIRECTLY WITH A SAVING IN TIME IF RCOND IS NOT NEEDED.
!
! ON ENTRY
!
! ABD REAL*8(LDA, N)
! CONTAINS THE MATRIX IN BAND STORAGE. THE COLUMNS
! OF THE MATRIX ARE STORED IN THE COLUMNS OF ABD AND
! THE DIAGONALS OF THE MATRIX ARE STORED IN ROWS
! ML+1 THROUGH 2*ML+MU+1 OF ABD .
! SEE THE COMMENTS BELOW FOR DETAILS.
!
! LDA INTEGER
! THE LEADING DIMENSION OF THE ARRAY ABD .
! LDA MUST BE .GE. 2*ML + MU + 1 .
!
! N INTEGER
! THE ORDER OF THE ORIGINAL MATRIX.
!
! ML INTEGER
! NUMBER OF DIAGONALS BELOW THE MAIN DIAGONAL.
! 0 .LE. ML .LT. N .
!
! MU INTEGER
! NUMBER OF DIAGONALS ABOVE THE MAIN DIAGONAL.
! 0 .LE. MU .LT. N .
! MORE EFFICIENT IF ML .LE. MU .
! ON RETURN
!
! ABD AN UPPER TRIANGULAR MATRIX IN BAND STORAGE AND
! THE MULTIPLIERS WHICH WERE USED TO OBTAIN IT.
! THE FACTORIZATION CAN BE WRITTEN A = L*U WHERE
! L IS A PRODUCT OF PERMUTATION AND UNIT LOWER
! TRIANGULAR MATRICES AND U IS UPPER TRIANGULAR.
!
! IPVT INTEGER(N)
! AN INTEGER VECTOR OF PIVOT INDICES.
!
! INFO INTEGER
! = 0 NORMAL VALUE.
! = K IF U(K,K) .EQ. 0.0 . THIS IS NOT AN ERROR
! CONDITION FOR THIS SUBROUTINE, BUT IT DOES
! INDICATE THAT DGBSL WILL DIVIDE BY ZERO IF
! CALLED. USE RCOND IN DGBCO FOR A RELIABLE
! INDICATION OF SINGULARITY.
!
! BAND STORAGE
!
! IF A IS A BAND MATRIX, THE FOLLOWING PROGRAM SEGMENT
! WILL SET UP THE INPUT.
!
! ML = (BAND WIDTH BELOW THE DIAGONAL)
! MU = (BAND WIDTH ABOVE THE DIAGONAL)
! M = ML + MU + 1
! DO 20 J = 1, N
! I1 = MAX0(1, J-MU)
! I2 = MIN0(N, J+ML)
! DO 10 I = I1, I2
! K = I - J + M
! ABD(K,J) = A(I,J)
! 10 CONTINUE
! 20 CONTINUE
!
! THIS USES ROWS ML+1 THROUGH 2*ML+MU+1 OF ABD .
! IN ADDITION, THE FIRST ML ROWS IN ABD ARE USED FOR
! ELEMENTS GENERATED DURING THE TRIANGULARIZATION.
! THE TOTAL NUMBER OF ROWS NEEDED IN ABD IS 2*ML+MU+1 .
! THE ML+MU BY ML+MU UPPER LEFT TRIANGLE AND THE
! ML BY ML LOWER RIGHT TRIANGLE ARE NOT REFERENCED.
!
! LINPACK. THIS VERSION DATED 08/14/78 .
! CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB.
!
! SUBROUTINES AND FUNCTIONS
!
! BLAS DAXPY,DSCAL,IDAMAX
! FORTRAN MAX0,MIN0
!
! INTERNAL VARIABLES
!
REAL*8 T
INTEGER I,I0,J,JU,JZ,J0,J1,K,KP1,L,LM,M,MM,NM1
!
!
M = ML + MU + 1
INFO = 0
!
! ZERO INITIAL FILL-IN COLUMNS
!
J0 = MU + 2
J1 = MIN0(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) = 0.0D0
10 CONTINUE
20 CONTINUE
30 CONTINUE
JZ = J1