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cdiag.f
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SUBROUTINE CDIAG(A,VALUE,VEC,N, NEED)
C
C TO FIND THE EIGENVALUES AND EIGENVECTORS OF A HERMITIAN MATRIX.
REAL VALUE(*),H
COMPLEX W(6000)
COMPLEX A(N,*),VEC(N,*)
COMPLEX FM06AS
IA=N
IV=N
C
C REDUCE MATRIX TO A TRI-DIAGONAL HERMITIAN MATRIX.
CALL ME08A(A,W,W(N+1),N,IA,W(2*N+1))
C
C FIND THE EIGENVALUES AND EIGENVECTORS OF THE TRI-DIAGONAL MATRIX
CALL EC08C(W,W(N+1),VALUE,VEC,N,IV,W(2*N+1))
IF(NEED.EQ.0) GOTO 50
IF(N.LT.2)RETURN
C
C THE EIGENVECTORS OF THE ORIGINAL MATRIX ARE NOW FOUND BY
C BACK TRANSFORMATION USING INFORMATION STORE IN THE UPPER
C TRIANGLE OF MATRIX A (BY ME08)
DO 40 II=3,N
I=N-II+1
H=W(N+I+1)*CONJG(A(I,I+1))
IF(H)10,40,10
10 DO30L=1,N
I1=I+1
S=FM06AS(N-I,A(I,I+1),IA,VEC(I+1,L),1)
S=S/H
DO20K=I1,N
20 VEC(K,L)=VEC(K,L)+CONJG(A(I,K))*S
30 CONTINUE
40 CONTINUE
50 CALL SORT(VALUE,VEC,N)
RETURN
END
SUBROUTINE EA08C(A,B,VALUE,VEC,M,IV,W)
C STANDARD FORTRAN 66 (A VERIFIED PFORT SUBROUTINE)
DIMENSION A(*),B(*),VALUE(*),VEC(*),W(*)
DATA EPS/1.E-6/,A34/0.0/
C THIS USES QR ITERATION TO FIND THE EIGENVALUES AND EIGENVECTORS
C OF THE SYMMETRIC TRIDIAGONAL MATRIX WHOSE DIAGONAL ELEMENTS ARE
C A(I),I=1,M AND OFF-DIAGONAL ELEMENTS ARE B(I),I=2,M. THE ARRAY
C W IS USED FOR WORKSPACE AND MUST HAVE DIMENSION AT LEAST 2*M.
C WE TREAT VEC AS IF IT HAD DIMENSIONS (IV,M).
SML=EPS*FLOAT(M)
CALL EA09C(A,B,W(M+1),M,W)
C SET VEC TO THE IDENTITY MATRIX.
DO 20 I=1,M
VALUE(I)=A(I)
W(I)=B(I)
K=(I-1)*IV+1
L=K+M-1
DO 10 J=K,L
10 VEC(J)=0.
KI=K+I-1
20 VEC(KI)=1.
K=0
IF(M.EQ.1)RETURN
DO 100 N3=2,M
N2=M+2-N3
C EACH QR ITERATION IS PERFORMED OF ROWS AND COLUMNS N1 TO N2
MN2=M+N2
ROOT=W(MN2)
DO 80 ITER=1,20
BB=(VALUE(N2)-VALUE(N2-1))*0.5
CC=W(N2)*W(N2)
A22=VALUE(N2)
IF(CC.NE.0.0)A22=A22+CC/(BB+SIGN(1.0,BB)*SQRT(BB*BB+CC))
DO 30 I=1,N2
MI=M+I
IF(ABS(ROOT-A22).LE.ABS(W(MI)-A22))GO TO 30
ROOT=W(MI)
MN=M+N2
W(MI)=W(MN)
W(MN)=ROOT
30 CONTINUE
DO 40 II=2,N2
N1=2+N2-II
IF(ABS(W(N1)).LE.(ABS(VALUE(N1-1))+ABS(VALUE(N1)))*SML)GO
1 TO 50
40 CONTINUE
N1=1
50 IF(N2.EQ.N1) GO TO 100
N2M1=N2-1
IF(ITER.GE.3)ROOT=A22
K=K+1
A22=VALUE(N1)
A12=A22-ROOT
A23=W(N1+1)
A13=A23
DO70 I=N1,N2M1
A33=VALUE(I+1)
IF(I.NE.N2M1)A34=W (I+2)
S=SIGN(SQRT(A12*A12+A13*A13),A12)
SI=A13/S
CO=A12/S
JK=I*IV+1
J1=JK-IV
J2=J1+MIN0(M,I+K)-1
DO 60 JI=J1,J2
V1=VEC(JI)
V2=VEC(JK)
VEC(JI)=V1*CO+V2*SI
VEC(JK)=V2*CO-V1*SI
60 JK=JK+1
IF(I.NE.N1) W(I)=S
A11=CO*A22+SI*A23
A12=CO*A23+SI*A33
A13=SI*A34
A21=CO*A23-SI*A22
A22=CO*A33-SI*A23
A23=CO*A34
VALUE(I)=A11*CO+A12*SI
A12=-A11*SI+A12*CO
W(I+1)=A12
70 A22=A22*CO-A21*SI
80 VALUE(N2)=A22
WRITE(6,90)
90 FORMAT(48H1CYCLE DETECTED IN SUBROUTINE EA08 -STOPPING NOW)
STOP
100 CONTINUE
C RAYLEIGH QUOTIENT
DO 120 J=1,M
K=(J-1)*IV
XX=VEC(K+1)**2
XAX=XX*A(1)
DO 110 I=2,M
KI=K+I
XX=XX+VEC(KI)**2
110 XAX=XAX+VEC(KI)*(2.*B(I)*VEC(KI-1)+A(I)*VEC(KI))
120 VALUE(J)=XAX/XX
RETURN
END
SUBROUTINE EA09C(A,B,VALUE,M,OFF)
C STANDARD FORTRAN 66 (A VERFIED PFORT SUBROUTINE)
DIMENSION A(*),B(*),VALUE(*),OFF(*)
DATA A34/0.0/,EPS/1.0E-6/
SML=EPS*FLOAT(M)
VALUE(1)=A(1)
IF(M.EQ.1)RETURN
DO 10 I=2,M
VALUE(I)=A(I)
10 OFF(I)=B(I)
C EACH QR ITERATION IS PERFORMED OF ROWS AND COLUMNS N1 TO N2
MAXIT=10*M
DO 80 ITER=1,MAXIT
DO 40 N3=2,M
N2=M+2-N3
DO 20 II=2,N2
N1=2+N2-II
IF(ABS(OFF(N1)).LE.(ABS(VALUE(N1-1))+ABS(VALUE(N1)))*SML)
1GO TO 30
20 CONTINUE
N1=1
30 IF(N2.NE.N1) GO TO 50
40 CONTINUE
RETURN
C ROOT IS THE EIGENVALUE OF THE BOTTOM 2*2 MATRIX THAT IS NEAREST
C TO THE LAST MATRIX ELEMENT AND IS USED TO ACCELERATE THE
C CONVERGENCE
50 BB=(VALUE(N2)-VALUE(N2-1))*0.5
CC=OFF(N2)*OFF(N2)
SBB=1.
IF(BB.LT.0.)SBB=-1.
ROOT=VALUE(N2)+CC/(BB+SBB*SQRT(BB*BB+CC))
N2M1=N2-1
60 A22=VALUE(N1)
A12=A22-ROOT
A23=OFF(N1+1)
A13=A23
DO 70 I=N1,N2M1
A33=VALUE(I+1)
IF(I.NE.N2M1)A34=OFF(I+2)
S=SQRT(A12*A12+A13*A13)
SI=A13/S
CO=A12/S
IF(I.NE.N1)OFF(I)=S
A11=CO*A22+SI*A23
A12=CO*A23+SI*A33
A13=SI*A34
A21=CO*A23-SI*A22
A22=CO*A33-SI*A23
A23=CO*A34
VALUE(I)=A11*CO+A12*SI
A12=-A11*SI+A12*CO
OFF(I+1)=A12
70 A22=A22*CO-A21*SI
80 VALUE(N2)=A22
WRITE(6,90)
90 FORMAT(39H1LOOPING DETECTED IN EA09-STOPPING NOW )
STOP
END
SUBROUTINE EC08C(A,B,VALUE,VEC,N,IV,W)
C
C TO FIND THE EIGENVALUES AND VECTORS OF A TRI-DIAGONAL
C HERMITIAN MATRIX.
REAL VALUE(*),W(*),PCK(2),ONE,ZERO,VEC(*)
COMPLEX A(*),B(*),DN,UPCK
EQUIVALENCE (PCK(1),UPCK)
C WE TREAT VEC AS IF IT IS DEFINED AS COMPLEX VEC(IV,N)
C IN THE CALLING PROGRAM.
DATA ONE, ZERO/1.0,0.0/
IV2=IV+IV
N2=N+N
IL=IV2*(N-1)+1
W(1)=A(1)
C
C THE HERMITIAN FORM IS TRANSFORMED INTO A REAL FORM
IF(N-1)80,80,10
10 DO 20 I=2,N
W(I)=A(I)
20 W(I+N)=CABS(B(I))
C
C FIND THE EIGENVALUES AND VECTORS OF THE REAL FORM
30 CALL EA08C(W,W(N+1),VALUE,VEC,N,IV2,W(N2+1))
C
C THE VECTORS IN VEC AT THIS POINT ARE REAL,WE NOW EXPAND THEM
C INTO VEC AS THOUGH THEY WERE COMPLEX.
DO 50 I=1,IL,IV2
DO 40 J=1,N
K=N-J
L=K+K
40 VEC(I+L)=VEC(I+K)
50 VEC(I+1)=ZERO
IF(N.LE.1)GO TO 80
DN=ONE
K=1
C
C TRANSFORM VECTORS OF REAL FORM TO THOSE OF COMPLEX FORM.
DO 70 I=4,N2,2
K=K+1
UPCK=ONE
IF(W(K+N).NE.ZERO)UPCK=DN*CONJG(B(K))/W(K+N)
I1=IL-1+I
DO 60 J=I,I1,IV2
VEC(J)=VEC(J-1)*PCK(2)
60 VEC(J-1)=VEC(J-1)*PCK(1)
70 DN=UPCK
80 RETURN
END
COMPLEX FUNCTION FM06AS(N,A,IA,B,IB)
IMPLICIT COMPLEX (A-H,O-Z)
COMPLEX A(*), B(*)
*******************************************************
*
* FM06AS - A FUNCTION ROUTINE TO COMPUTE THE VALUE OF THE
* INNER PRODUCT, OR DOT PRODUCT, OF TWO SINGLE PRECISION
* COMPLEX VECTORS, ACCUMULATING THE INTERMEDIATE PRODUCTS
* DOUBLE PRECISION. THE ELEMENTS OF EACH VECTOR CAN BE
* STORED IN ANY FIXED DISPLACEMENT FROM NEIGHBOURING
* ELEMENTS.
*
* COMPUTES: SUM(J=1,N) A((J-1)*IA+1)*B((J-1)*IB+1)
*
* W = FM06AS(N,A,IA,B,IB)
*
* N INTEGER SCALAR; (USER:*); LENGTH OF THE VECTORS A AND B.
* IF N <= 0 THE INNER PRODUCT VALUE IS DEFINED TO BE ZERO.
* A COMPLEX*8 ARRAY((N-1)*IABS(IA)+1); (USER:*); THE ARRAY
* CONTAINING THE 1ST VECTOR. THE FORTRAN CONVENTION OF STORING
* REAL AND IMAGINARY PARTS IN ADJACENT WORDS IS ASSUMED.
* IA INTEGER SCALAR; (USER:*); THE SUBSCRIPT DISPLACEMENT OF
* AN ELEMENT IN THE ARRAY A TO ITS NEIGHBOUR, I.E. THE VECTOR
* ELEMENTS ARE IN A(1), A(IA+1), A(2*IA+1),...
* IF IA < 0 THE ELEMENTS ARE ASSUMED TO BE STORED IN
* A(1-(N-1)*IA), A(1-(N-2)*IA),..., A(1-IA), A(1).
* B COMPLEX*8 ARRAY((N-1)*IABS(IA)+1); (USER:*); THE ARRAY
* CONTAINING THE SECOND VECTOR. TREAT LIKE A.
* IB INTEGER SCALAR; (USER:*); THE SUBSCRIPT DISPLACEMENT OF
* AN ELEMENT IN B TO ITS NEIGHBOUR. TREAT LIKE IA.
* FM06AS COMPLEX FUNCTION; (*:FM06AS); THE INNER PRODUCT VALUE.
* IT IS RETURNED DOUBLE PRECISION, THE REAL PART IN FLT PNT
* REG 0 AND THE IMAGINARY PART IN FLT PNT REG 2.
*
* THIS ROUTINE IS WRITTEN IN FORTRAN.
*
*--------------------------------------------------------*
SUM=(0.0,0.0)
DO 10 I=1,N
10 SUM=SUM+A((I-1)*IA+1)*B((I-1)*IB+1)
FM06AS=SUM
RETURN
END
COMPLEX FUNCTION FM06BS(N,A,IA,B,IB)
IMPLICIT COMPLEX (A-H,O-Z)
COMPLEX A(*), B(*)
*******************************************************
*
* FM06BS - A FUNCTION ROUTINE TO COMPUTE THE VALUE OF THE
* INNER PRODUCT, OR DOT PRODUCT, OF A SIGLE PRECISION
* COMPLEX VECTORS, ACCUMULATING THE INTERMEDIATE PRODUCTS
* DOUBLE PRECISION. THE ELEMENTS OF EACH VECTOR CAN BE
* STORED IN ANY FIXED DISPLACEMENT FROM NEIGHBOURING
* ELEMENTS.
*
* COMPUTES: SUM(J=1,N) A((J-1)*IA+1)*B((J-1)*IB+1)
*
* W = FM06BS(N,A,IA,B,IB)
*
* N INTEGER SCALAR; (USER:*); LENGTH OF THE VECTORS A AND B.
* IF N <= 0 THE INNER PRODUCT VALUE IS DEFINED TO BE ZERO.
* A COMPLEX*8 ARRAY((N-1)*IABS(IA)+1); (USER:*); THE ARRAY
* CONTAINING THE 1ST VECTOR. THE FORTRAN CONVENTION OF STORING
* REAL AND IMAGINARY PARTS IN ADJACENT WORDS IS ASSUMED.
* IA INTEGER SCALAR; (USER:*); THE SUBSCRIPT DISPLACEMENT OF
* AN ELEMENT IN THE ARRAY A TO ITS NEIGHBOUR, I.E. THE VECTOR
* ELEMENTS ARE IN A(1), A(IA+1), A(2*IA+1),...
* IF IA < 0 THE ELEMENTS ARE ASSUMED TO BE STORED IN
* A(1-(N-1)*IA), A(1-(N-2)*IA),..., A(1-IA), A(1).
* B COMPLEX*8 ARRAY((N-1)*IABS(IA)+1); (USER:*); THE ARRAY
* CONTAINING THE SECOND VECTOR. TREAT LIKE A.
* IB INTEGER SCALAR; (USER:*); THE SUBSCRIPT DISPLACEMENT OF
* AN ELEMENT IN B TO ITS NEIGHBOUR. TREAT LIKE IA.
* FM06BS COMPLEX FUNCTION; (*:FM06BS); THE INNER PRODUCT VALUE.
* IT IS RETURNED DOUBLE PRECISION, THE REAL PART IN FLT PNT
* REG 0 AND THE IMAGINARY PART IN FLT PNT REG 2.
*
* THIS ROUTINE IS WRITTEN IN FORTRAN.
*
*--------------------------------------------------------*
SUM=(0.0,0.0)
DO 10 I=1,N
10 SUM=SUM+A((I-1)*IA+1)*CONJG(B((I-1)*IB+1))
FM06BS=SUM
RETURN
END
SUBROUTINE ME08A(A,ALPHA,BETA,N,IA,Q)
COMPLEX A(IA,*),ALPHA(*),BETA(*),Q(*),CW,QJ
COMPLEX FM06AS,FM06BS
REAL PP,ZERO,P5,H
REAL S1,PP1
DATA ZERO/0.0/, P5 /0.50/
C**************************************************************
C THE REDUCTION OF FULL HERMITIAN MATRIX INTO TRI-DIAGONAL HERMITIAN
C FORM IS DONE IN N-2 STEPS.AT THE I TH STEP ZEROS ARE INTRODUCED IN
C THE I TH ROW AND COLUMNS WITHOUT DESTROYING PREVIOUSLY PRODUCED ZEROS
C
C H
C AT THE I TH STEP WE HAVE A =P A P WITH P =I-U U /K
C I I I-1 I I I I I
C
C WHERE U =(0,0,...,A (1+S /T ),A ,...,A )
C I I,I+1 I I I,I+2 I,N
C 2 N 2 2 2
C S = SUM ] A ] K =S +T AND T = SQRT(]A ] S )
C I J=I+1 I,J I I I I I,I+1 I
C
C COMPUTATIONAL DETAILS AT THE I TH STAGE ARE (1) FORM S ,K THEN
C I I
C
C H H H
C (2) Q =A U /K (3) Q =Q -.5U (U Q /K ) (4) A =A -U Q -Q U
C I I-1 I I I I I I I I I I-1 I I I I
C
C THE VECTORS U BEING APPROPIATELY IN A.
IF(N.LE.0)GO TO 90
DO 10 J=1,N
ALPHA(J)=A(J,J)
DO 10 I=1,J
10 A(I,J)=CONJG(A(J,I))
IF(N-2)90,80,20
20 N2=N-2
DO 60 I=1,N2
I1=I+1
C (1)
CW=FM06BS(N-I,A(I,I+1),IA,A(I,I+1),IA)
PP=CW
PP1=SQRT(PP)
BETA(I+1)=CMPLX(-PP1,ZERO)
S1=CABS(A(I,I+1))
IF(S1.GT.ZERO)BETA(I+1)=BETA(I+1)*A(I,I+1)/S1
IF(PP.LE.1.D-15)GO TO 60
H=PP+PP1*S1
A(I,I+1)=A(I,I+1)-BETA(I+1)
C (2)
DO 30 K=I1,N
QJ=FM06AS(-(I-K),A(I+1,K),1,A(I,I+1),IA)
QJ=FM06BS(N-K,A(K,K+1),IA,A(I,K+1),IA)+CONJG(QJ)
30 Q(K)=QJ/H
C (3)
CW=FM06AS(N-I,A(I,I+1),IA,Q(I+1),1)
PP=CW*P5/H
DO 40 K=I1,N
40 Q(K)=Q(K)-PP*CONJG(A(I,K))
C (4)
DO 50 K=I1,N
50 CALL ME08B (A(K,K),Q(K),A(I,K),N-K+1,IA*2)
60 CONTINUE
DO 70 I=2,N
QJ=ALPHA(I)
ALPHA(I)=A(I,I)
70 A(I,I)=QJ
80 BETA(N)=A(N-1,N)
90 RETURN
END
SUBROUTINE ME08B (A,Q,B,N,IA)
REAL A(IA,*),Q(2,*),B(IA,*)
DO 10 I=1,N
A(1,I)=A(1,I) -Q(1,1)*B(1,I)+Q(2,1)*B(2,I)
1 -Q(1,I)*B(1,1)+Q(2,I)*B(2,1)
10 A(2,I)=A(2,I)-Q(2,1)*B(1,I)-Q(1,1)*B(2,I)
1 +Q(2,I)*B(1,1)+Q(1,I)*B(2,1)
RETURN
END
SUBROUTINE SORT(VAL,VEC,N)
COMPLEX VEC(N,*), SUM
REAL VAL(*)
DO 30 I=1,N
X=1.E9
DO 10 J=I,N
IF(VAL(J).LT.X) THEN
K=J
X=VAL(J)
ENDIF
10 CONTINUE
DO 20 J=1,N
SUM=VEC(J,K)
VEC(J,K)=VEC(J,I)
20 VEC(J,I)=SUM
VAL(K)=VAL(I)
VAL(I)=X
30 CONTINUE
RETURN
END