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eigen.f
3496 lines (3496 loc) · 103 KB
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eigen.f
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SUBROUTINE BALANC(NM,N,A,LOW,IGH,SCALE)
C
INTEGER I,J,K,L,M,N,JJ,NM,IGH,LOW,IEXC
DOUBLE PRECISION A(NM,N),SCALE(N)
DOUBLE PRECISION C,F,G,R,S,B2,RADIX
LOGICAL NOCONV
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE BALANCE,
C NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C THIS SUBROUTINE BALANCES A REAL MATRIX AND ISOLATES
C EIGENVALUES WHENEVER POSSIBLE.
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 A CONTAINS THE INPUT MATRIX TO BE BALANCED.
C
C ON OUTPUT
C
C A CONTAINS THE BALANCED MATRIX.
C
C LOW AND IGH ARE TWO INTEGERS SUCH THAT A(I,J)
C IS EQUAL TO ZERO IF
C (1) I IS GREATER THAN J AND
C (2) J=1,...,LOW-1 OR I=IGH+1,...,N.
C
C SCALE CONTAINS INFORMATION DETERMINING THE
C PERMUTATIONS AND SCALING FACTORS USED.
C
C SUPPOSE THAT THE PRINCIPAL SUBMATRIX IN ROWS LOW THROUGH IGH
C HAS BEEN BALANCED, THAT P(J) DENOTES THE INDEX INTERCHANGED
C WITH J DURING THE PERMUTATION STEP, AND THAT THE ELEMENTS
C OF THE DIAGONAL MATRIX USED ARE DENOTED BY D(I,J). THEN
C SCALE(J) = P(J), FOR J = 1,...,LOW-1
C = D(J,J), J = LOW,...,IGH
C = P(J) J = IGH+1,...,N.
C THE ORDER IN WHICH THE INTERCHANGES ARE MADE IS N TO IGH+1,
C THEN 1 TO LOW-1.
C
C NOTE THAT 1 IS RETURNED FOR IGH IF IGH IS ZERO FORMALLY.
C
C THE ALGOL PROCEDURE EXC CONTAINED IN BALANCE APPEARS IN
C BALANC IN LINE. (NOTE THAT THE ALGOL ROLES OF IDENTIFIERS
C K,L HAVE BEEN REVERSED.)
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
C
RADIX = 16.0D0
C
B2 = RADIX * RADIX
K = 1
L = N
GO TO 100
C .......... IN-LINE PROCEDURE FOR ROW AND
C COLUMN EXCHANGE ..........
20 SCALE(M) = J
IF (J .EQ. M) GO TO 50
C
DO 30 I = 1, L
F = A(I,J)
A(I,J) = A(I,M)
A(I,M) = F
30 CONTINUE
C
DO 40 I = K, N
F = A(J,I)
A(J,I) = A(M,I)
A(M,I) = F
40 CONTINUE
C
50 GO TO (80,130), IEXC
C .......... SEARCH FOR ROWS ISOLATING AN EIGENVALUE
C AND PUSH THEM DOWN ..........
80 IF (L .EQ. 1) GO TO 280
L = L - 1
C .......... FOR J=L STEP -1 UNTIL 1 DO -- ..........
100 DO 120 JJ = 1, L
J = L + 1 - JJ
C
DO 110 I = 1, L
IF (I .EQ. J) GO TO 110
IF (A(J,I) .NE. 0.0D0) GO TO 120
110 CONTINUE
C
M = L
IEXC = 1
GO TO 20
120 CONTINUE
C
GO TO 140
C .......... SEARCH FOR COLUMNS ISOLATING AN EIGENVALUE
C AND PUSH THEM LEFT ..........
130 K = K + 1
C
140 DO 170 J = K, L
C
DO 150 I = K, L
IF (I .EQ. J) GO TO 150
IF (A(I,J) .NE. 0.0D0) GO TO 170
150 CONTINUE
C
M = K
IEXC = 2
GO TO 20
170 CONTINUE
C .......... NOW BALANCE THE SUBMATRIX IN ROWS K TO L ..........
DO 180 I = K, L
180 SCALE(I) = 1.0D0
C .......... ITERATIVE LOOP FOR NORM REDUCTION ..........
190 NOCONV = .FALSE.
C
DO 270 I = K, L
C = 0.0D0
R = 0.0D0
C
DO 200 J = K, L
IF (J .EQ. I) GO TO 200
C = C + DABS(A(J,I))
R = R + DABS(A(I,J))
200 CONTINUE
C .......... GUARD AGAINST ZERO C OR R DUE TO UNDERFLOW ..........
IF (C .EQ. 0.0D0 .OR. R .EQ. 0.0D0) GO TO 270
G = R / RADIX
F = 1.0D0
S = C + R
210 IF (C .GE. G) GO TO 220
F = F * RADIX
C = C * B2
GO TO 210
220 G = R * RADIX
230 IF (C .LT. G) GO TO 240
F = F / RADIX
C = C / B2
GO TO 230
C .......... NOW BALANCE ..........
240 IF ((C + R) / F .GE. 0.95D0 * S) GO TO 270
G = 1.0D0 / F
SCALE(I) = SCALE(I) * F
NOCONV = .TRUE.
C
DO 250 J = K, N
250 A(I,J) = A(I,J) * G
C
DO 260 J = 1, L
260 A(J,I) = A(J,I) * F
C
270 CONTINUE
C
IF (NOCONV) GO TO 190
C
280 LOW = K
IGH = L
RETURN
END
SUBROUTINE BALBAK(NM,N,LOW,IGH,SCALE,M,Z)
C
INTEGER I,J,K,M,N,II,NM,IGH,LOW
DOUBLE PRECISION SCALE(N),Z(NM,M)
DOUBLE PRECISION S
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE BALBAK,
C NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C THIS SUBROUTINE FORMS THE EIGENVECTORS OF A REAL GENERAL
C MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
C BALANCED MATRIX DETERMINED BY BALANC.
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 BALANC.
C
C SCALE CONTAINS INFORMATION DETERMINING THE PERMUTATIONS
C AND SCALING FACTORS USED BY BALANC.
C
C M IS THE NUMBER OF COLUMNS OF Z TO BE BACK TRANSFORMED.
C
C Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE EIGEN-
C VECTORS TO BE BACK TRANSFORMED IN ITS FIRST M COLUMNS.
C
C ON OUTPUT
C
C Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE
C TRANSFORMED EIGENVECTORS IN ITS FIRST M COLUMNS.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
C
IF (M .EQ. 0) GO TO 200
IF (IGH .EQ. LOW) GO TO 120
C
DO 110 I = LOW, IGH
S = SCALE(I)
C .......... LEFT HAND EIGENVECTORS ARE BACK TRANSFORMED
C IF THE FOREGOING STATEMENT IS REPLACED BY
C S=1.0D0/SCALE(I). ..........
DO 100 J = 1, M
100 Z(I,J) = Z(I,J) * S
C
110 CONTINUE
C ......... FOR I=LOW-1 STEP -1 UNTIL 1,
C IGH+1 STEP 1 UNTIL N DO -- ..........
120 DO 140 II = 1, N
I = II
IF (I .GE. LOW .AND. I .LE. IGH) GO TO 140
IF (I .LT. LOW) I = LOW - II
K = SCALE(I)
IF (K .EQ. I) GO TO 140
C
DO 130 J = 1, M
S = Z(I,J)
Z(I,J) = Z(K,J)
Z(K,J) = S
130 CONTINUE
C
140 CONTINUE
C
200 RETURN
END
SUBROUTINE CBABK2(NM,N,LOW,IGH,SCALE,M,ZR,ZI)
C
INTEGER I,J,K,M,N,II,NM,IGH,LOW
DOUBLE PRECISION SCALE(N),ZR(NM,M),ZI(NM,M)
DOUBLE PRECISION S
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE
C CBABK2, WHICH IS A COMPLEX VERSION OF BALBAK,
C NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C THIS SUBROUTINE FORMS THE EIGENVECTORS OF A COMPLEX GENERAL
C MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
C BALANCED MATRIX DETERMINED BY CBAL.
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 CBAL.
C
C SCALE CONTAINS INFORMATION DETERMINING THE PERMUTATIONS
C AND SCALING FACTORS USED BY CBAL.
C
C M IS THE NUMBER OF EIGENVECTORS TO BE BACK TRANSFORMED.
C
C ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE EIGENVECTORS TO BE
C BACK TRANSFORMED IN THEIR FIRST M COLUMNS.
C
C ON OUTPUT
C
C ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE TRANSFORMED EIGENVECTORS
C IN THEIR FIRST M COLUMNS.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
C
IF (M .EQ. 0) GO TO 200
IF (IGH .EQ. LOW) GO TO 120
C
DO 110 I = LOW, IGH
S = SCALE(I)
C .......... LEFT HAND EIGENVECTORS ARE BACK TRANSFORMED
C IF THE FOREGOING STATEMENT IS REPLACED BY
C S=1.0D0/SCALE(I). ..........
DO 100 J = 1, M
ZR(I,J) = ZR(I,J) * S
ZI(I,J) = ZI(I,J) * S
100 CONTINUE
C
110 CONTINUE
C .......... FOR I=LOW-1 STEP -1 UNTIL 1,
C IGH+1 STEP 1 UNTIL N DO -- ..........
120 DO 140 II = 1, N
I = II
IF (I .GE. LOW .AND. I .LE. IGH) GO TO 140
IF (I .LT. LOW) I = LOW - II
K = SCALE(I)
IF (K .EQ. I) GO TO 140
C
DO 130 J = 1, M
S = ZR(I,J)
ZR(I,J) = ZR(K,J)
ZR(K,J) = S
S = ZI(I,J)
ZI(I,J) = ZI(K,J)
ZI(K,J) = S
130 CONTINUE
C
140 CONTINUE
C
200 RETURN
END
SUBROUTINE CBAL(NM,N,AR,AI,LOW,IGH,SCALE)
C
INTEGER I,J,K,L,M,N,JJ,NM,IGH,LOW,IEXC
DOUBLE PRECISION AR(NM,N),AI(NM,N),SCALE(N)
DOUBLE PRECISION C,F,G,R,S,B2,RADIX
LOGICAL NOCONV
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE
C CBALANCE, WHICH IS A COMPLEX VERSION OF BALANCE,
C NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C THIS SUBROUTINE BALANCES A COMPLEX MATRIX AND ISOLATES
C EIGENVALUES WHENEVER POSSIBLE.
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 AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE COMPLEX MATRIX TO BE BALANCED.
C
C ON OUTPUT
C
C AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE BALANCED MATRIX.
C
C LOW AND IGH ARE TWO INTEGERS SUCH THAT AR(I,J) AND AI(I,J)
C ARE EQUAL TO ZERO IF
C (1) I IS GREATER THAN J AND
C (2) J=1,...,LOW-1 OR I=IGH+1,...,N.
C
C SCALE CONTAINS INFORMATION DETERMINING THE
C PERMUTATIONS AND SCALING FACTORS USED.
C
C SUPPOSE THAT THE PRINCIPAL SUBMATRIX IN ROWS LOW THROUGH IGH
C HAS BEEN BALANCED, THAT P(J) DENOTES THE INDEX INTERCHANGED
C WITH J DURING THE PERMUTATION STEP, AND THAT THE ELEMENTS
C OF THE DIAGONAL MATRIX USED ARE DENOTED BY D(I,J). THEN
C SCALE(J) = P(J), FOR J = 1,...,LOW-1
C = D(J,J) J = LOW,...,IGH
C = P(J) J = IGH+1,...,N.
C THE ORDER IN WHICH THE INTERCHANGES ARE MADE IS N TO IGH+1,
C THEN 1 TO LOW-1.
C
C NOTE THAT 1 IS RETURNED FOR IGH IF IGH IS ZERO FORMALLY.
C
C THE ALGOL PROCEDURE EXC CONTAINED IN CBALANCE APPEARS IN
C CBAL IN LINE. (NOTE THAT THE ALGOL ROLES OF IDENTIFIERS
C K,L HAVE BEEN REVERSED.)
C
C ARITHMETIC IS REAL THROUGHOUT.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
C
RADIX = 16.0D0
C
B2 = RADIX * RADIX
K = 1
L = N
GO TO 100
C .......... IN-LINE PROCEDURE FOR ROW AND
C COLUMN EXCHANGE ..........
20 SCALE(M) = J
IF (J .EQ. M) GO TO 50
C
DO 30 I = 1, L
F = AR(I,J)
AR(I,J) = AR(I,M)
AR(I,M) = F
F = AI(I,J)
AI(I,J) = AI(I,M)
AI(I,M) = F
30 CONTINUE
C
DO 40 I = K, N
F = AR(J,I)
AR(J,I) = AR(M,I)
AR(M,I) = F
F = AI(J,I)
AI(J,I) = AI(M,I)
AI(M,I) = F
40 CONTINUE
C
50 GO TO (80,130), IEXC
C .......... SEARCH FOR ROWS ISOLATING AN EIGENVALUE
C AND PUSH THEM DOWN ..........
80 IF (L .EQ. 1) GO TO 280
L = L - 1
C .......... FOR J=L STEP -1 UNTIL 1 DO -- ..........
100 DO 120 JJ = 1, L
J = L + 1 - JJ
C
DO 110 I = 1, L
IF (I .EQ. J) GO TO 110
IF (AR(J,I) .NE. 0.0D0 .OR. AI(J,I) .NE. 0.0D0) GO TO 120
110 CONTINUE
C
M = L
IEXC = 1
GO TO 20
120 CONTINUE
C
GO TO 140
C .......... SEARCH FOR COLUMNS ISOLATING AN EIGENVALUE
C AND PUSH THEM LEFT ..........
130 K = K + 1
C
140 DO 170 J = K, L
C
DO 150 I = K, L
IF (I .EQ. J) GO TO 150
IF (AR(I,J) .NE. 0.0D0 .OR. AI(I,J) .NE. 0.0D0) GO TO 170
150 CONTINUE
C
M = K
IEXC = 2
GO TO 20
170 CONTINUE
C .......... NOW BALANCE THE SUBMATRIX IN ROWS K TO L ..........
DO 180 I = K, L
180 SCALE(I) = 1.0D0
C .......... ITERATIVE LOOP FOR NORM REDUCTION ..........
190 NOCONV = .FALSE.
C
DO 270 I = K, L
C = 0.0D0
R = 0.0D0
C
DO 200 J = K, L
IF (J .EQ. I) GO TO 200
C = C + DABS(AR(J,I)) + DABS(AI(J,I))
R = R + DABS(AR(I,J)) + DABS(AI(I,J))
200 CONTINUE
C .......... GUARD AGAINST ZERO C OR R DUE TO UNDERFLOW ..........
IF (C .EQ. 0.0D0 .OR. R .EQ. 0.0D0) GO TO 270
G = R / RADIX
F = 1.0D0
S = C + R
210 IF (C .GE. G) GO TO 220
F = F * RADIX
C = C * B2
GO TO 210
220 G = R * RADIX
230 IF (C .LT. G) GO TO 240
F = F / RADIX
C = C / B2
GO TO 230
C .......... NOW BALANCE ..........
240 IF ((C + R) / F .GE. 0.95D0 * S) GO TO 270
G = 1.0D0 / F
SCALE(I) = SCALE(I) * F
NOCONV = .TRUE.
C
DO 250 J = K, N
AR(I,J) = AR(I,J) * G
AI(I,J) = AI(I,J) * G
250 CONTINUE
C
DO 260 J = 1, L
AR(J,I) = AR(J,I) * F
AI(J,I) = AI(J,I) * F
260 CONTINUE
C
270 CONTINUE
C
IF (NOCONV) GO TO 190
C
280 LOW = K
IGH = L
RETURN
END
SUBROUTINE CDIV(AR,AI,BR,BI,CR,CI)
DOUBLE PRECISION AR,AI,BR,BI,CR,CI
C
C COMPLEX DIVISION, (CR,CI) = (AR,AI)/(BR,BI)
C
DOUBLE PRECISION S,ARS,AIS,BRS,BIS
S = DABS(BR) + DABS(BI)
ARS = AR/S
AIS = AI/S
BRS = BR/S
BIS = BI/S
S = BRS**2 + BIS**2
CR = (ARS*BRS + AIS*BIS)/S
CI = (AIS*BRS - ARS*BIS)/S
RETURN
END
SUBROUTINE COMQR(NM,N,LOW,IGH,HR,HI,WR,WI,IERR)
C
INTEGER I,J,L,N,EN,LL,NM,IGH,ITN,ITS,LOW,LP1,ENM1,IERR
DOUBLE PRECISION HR(NM,N),HI(NM,N),WR(N),WI(N)
DOUBLE PRECISION SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,TST1,TST2,
X PYTHAG
C
C THIS SUBROUTINE IS A TRANSLATION OF A UNITARY ANALOGUE OF THE
C ALGOL PROCEDURE COMLR, NUM. MATH. 12, 369-376(1968) BY MARTIN
C AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 396-403(1971).
C THE UNITARY ANALOGUE SUBSTITUTES THE QR ALGORITHM OF FRANCIS
C (COMP. JOUR. 4, 332-345(1962)) FOR THE LR ALGORITHM.
C
C THIS SUBROUTINE FINDS THE EIGENVALUES OF A COMPLEX
C UPPER HESSENBERG MATRIX BY THE QR METHOD.
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 CBAL. IF CBAL HAS NOT BEEN USED,
C SET LOW=1, IGH=N.
C
C HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX.
C THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN
C INFORMATION ABOUT THE UNITARY TRANSFORMATIONS USED IN
C THE REDUCTION BY CORTH, IF PERFORMED.
C
C ON OUTPUT
C
C THE UPPER HESSENBERG PORTIONS OF HR AND HI HAVE BEEN
C DESTROYED. THEREFORE, THEY MUST BE SAVED BEFORE
C CALLING COMQR IF SUBSEQUENT CALCULATION OF
C EIGENVECTORS IS TO BE PERFORMED.
C
C WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE EIGENVALUES. IF AN ERROR
C EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT
C FOR INDICES IERR+1,...,N.
C
C IERR IS SET TO
C ZERO FOR NORMAL RETURN,
C J IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED
C WHILE THE J-TH EIGENVALUE IS BEING SOUGHT.
C
C CALLS CDIV FOR COMPLEX DIVISION.
C CALLS CSROOT FOR COMPLEX SQUARE ROOT.
C CALLS PYTHAG FOR DSQRT(A*A + B*B) .
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
c
c unnecessary initialization of L to keep g77 -Wall happy
c
L = 0
C
IERR = 0
IF (LOW .EQ. IGH) GO TO 180
C .......... CREATE REAL SUBDIAGONAL ELEMENTS ..........
L = LOW + 1
C
DO 170 I = L, IGH
LL = MIN0(I+1,IGH)
IF (HI(I,I-1) .EQ. 0.0D0) GO TO 170
NORM = PYTHAG(HR(I,I-1),HI(I,I-1))
YR = HR(I,I-1) / NORM
YI = HI(I,I-1) / NORM
HR(I,I-1) = NORM
HI(I,I-1) = 0.0D0
C
DO 155 J = I, IGH
SI = YR * HI(I,J) - YI * HR(I,J)
HR(I,J) = YR * HR(I,J) + YI * HI(I,J)
HI(I,J) = SI
155 CONTINUE
C
DO 160 J = LOW, LL
SI = YR * HI(J,I) + YI * HR(J,I)
HR(J,I) = YR * HR(J,I) - YI * HI(J,I)
HI(J,I) = SI
160 CONTINUE
C
170 CONTINUE
C .......... STORE ROOTS ISOLATED BY CBAL ..........
180 DO 200 I = 1, N
IF (I .GE. LOW .AND. I .LE. IGH) GO TO 200
WR(I) = HR(I,I)
WI(I) = HI(I,I)
200 CONTINUE
C
EN = IGH
TR = 0.0D0
TI = 0.0D0
ITN = 30*N
C .......... SEARCH FOR NEXT EIGENVALUE ..........
220 IF (EN .LT. LOW) GO TO 1001
ITS = 0
ENM1 = EN - 1
C .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT
C FOR L=EN STEP -1 UNTIL LOW D0 -- ..........
240 DO 260 LL = LOW, EN
L = EN + LOW - LL
IF (L .EQ. LOW) GO TO 300
TST1 = DABS(HR(L-1,L-1)) + DABS(HI(L-1,L-1))
X + DABS(HR(L,L)) + DABS(HI(L,L))
TST2 = TST1 + DABS(HR(L,L-1))
IF (TST2 .EQ. TST1) GO TO 300
260 CONTINUE
C .......... FORM SHIFT ..........
300 IF (L .EQ. EN) GO TO 660
IF (ITN .EQ. 0) GO TO 1000
IF (ITS .EQ. 10 .OR. ITS .EQ. 20) GO TO 320
SR = HR(EN,EN)
SI = HI(EN,EN)
XR = HR(ENM1,EN) * HR(EN,ENM1)
XI = HI(ENM1,EN) * HR(EN,ENM1)
IF (XR .EQ. 0.0D0 .AND. XI .EQ. 0.0D0) GO TO 340
YR = (HR(ENM1,ENM1) - SR) / 2.0D0
YI = (HI(ENM1,ENM1) - SI) / 2.0D0
CALL CSROOT(YR**2-YI**2+XR,2.0D0*YR*YI+XI,ZZR,ZZI)
IF (YR * ZZR + YI * ZZI .GE. 0.0D0) GO TO 310
ZZR = -ZZR
ZZI = -ZZI
310 CALL CDIV(XR,XI,YR+ZZR,YI+ZZI,XR,XI)
SR = SR - XR
SI = SI - XI
GO TO 340
C .......... FORM EXCEPTIONAL SHIFT ..........
320 SR = DABS(HR(EN,ENM1)) + DABS(HR(ENM1,EN-2))
SI = 0.0D0
C
340 DO 360 I = LOW, EN
HR(I,I) = HR(I,I) - SR
HI(I,I) = HI(I,I) - SI
360 CONTINUE
C
TR = TR + SR
TI = TI + SI
ITS = ITS + 1
ITN = ITN - 1
C .......... REDUCE TO TRIANGLE (ROWS) ..........
LP1 = L + 1
C
DO 500 I = LP1, EN
SR = HR(I,I-1)
HR(I,I-1) = 0.0D0
NORM = PYTHAG(PYTHAG(HR(I-1,I-1),HI(I-1,I-1)),SR)
XR = HR(I-1,I-1) / NORM
WR(I-1) = XR
XI = HI(I-1,I-1) / NORM
WI(I-1) = XI
HR(I-1,I-1) = NORM
HI(I-1,I-1) = 0.0D0
HI(I,I-1) = SR / NORM
C
DO 490 J = I, EN
YR = HR(I-1,J)
YI = HI(I-1,J)
ZZR = HR(I,J)
ZZI = HI(I,J)
HR(I-1,J) = XR * YR + XI * YI + HI(I,I-1) * ZZR
HI(I-1,J) = XR * YI - XI * YR + HI(I,I-1) * ZZI
HR(I,J) = XR * ZZR - XI * ZZI - HI(I,I-1) * YR
HI(I,J) = XR * ZZI + XI * ZZR - HI(I,I-1) * YI
490 CONTINUE
C
500 CONTINUE
C
SI = HI(EN,EN)
IF (SI .EQ. 0.0D0) GO TO 540
NORM = PYTHAG(HR(EN,EN),SI)
SR = HR(EN,EN) / NORM
SI = SI / NORM
HR(EN,EN) = NORM
HI(EN,EN) = 0.0D0
C .......... INVERSE OPERATION (COLUMNS) ..........
540 DO 600 J = LP1, EN
XR = WR(J-1)
XI = WI(J-1)
C
DO 580 I = L, J
YR = HR(I,J-1)
YI = 0.0D0
ZZR = HR(I,J)
ZZI = HI(I,J)
IF (I .EQ. J) GO TO 560
YI = HI(I,J-1)
HI(I,J-1) = XR * YI + XI * YR + HI(J,J-1) * ZZI
560 HR(I,J-1) = XR * YR - XI * YI + HI(J,J-1) * ZZR
HR(I,J) = XR * ZZR + XI * ZZI - HI(J,J-1) * YR
HI(I,J) = XR * ZZI - XI * ZZR - HI(J,J-1) * YI
580 CONTINUE
C
600 CONTINUE
C
IF (SI .EQ. 0.0D0) GO TO 240
C
DO 630 I = L, EN
YR = HR(I,EN)
YI = HI(I,EN)
HR(I,EN) = SR * YR - SI * YI
HI(I,EN) = SR * YI + SI * YR
630 CONTINUE
C
GO TO 240
C .......... A ROOT FOUND ..........
660 WR(EN) = HR(EN,EN) + TR
WI(EN) = HI(EN,EN) + TI
EN = ENM1
GO TO 220
C .......... SET ERROR -- ALL EIGENVALUES HAVE NOT
C CONVERGED AFTER 30*N ITERATIONS ..........
1000 IERR = EN
1001 RETURN
END
SUBROUTINE COMQR2(NM,N,LOW,IGH,ORTR,ORTI,HR,HI,WR,WI,ZR,ZI,IERR)
C
INTEGER I,J,K,L,M,N,EN,II,JJ,LL,NM,NN,IGH,IP1,
X ITN,ITS,LOW,LP1,ENM1,IEND,IERR
DOUBLE PRECISION HR(NM,N),HI(NM,N),WR(N),WI(N),ZR(NM,N),ZI(NM,N),
X ORTR(IGH),ORTI(IGH)
DOUBLE PRECISION SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,TST1,TST2,
X PYTHAG
C
C THIS SUBROUTINE IS A TRANSLATION OF A UNITARY ANALOGUE OF THE
C ALGOL PROCEDURE COMLR2, NUM. MATH. 16, 181-204(1970) BY PETERS
C AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
C THE UNITARY ANALOGUE SUBSTITUTES THE QR ALGORITHM OF FRANCIS
C (COMP. JOUR. 4, 332-345(1962)) FOR THE LR ALGORITHM.
C
C THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS
C OF A COMPLEX UPPER HESSENBERG MATRIX BY THE QR
C METHOD. THE EIGENVECTORS OF A COMPLEX GENERAL MATRIX
C CAN ALSO BE FOUND IF CORTH HAS BEEN USED TO REDUCE
C THIS GENERAL MATRIX TO HESSENBERG FORM.
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 CBAL. IF CBAL HAS NOT BEEN USED,
C SET LOW=1, IGH=N.
C
C ORTR AND ORTI CONTAIN INFORMATION ABOUT THE UNITARY TRANS-
C FORMATIONS USED IN THE REDUCTION BY CORTH, IF PERFORMED.
C ONLY ELEMENTS LOW THROUGH IGH ARE USED. IF THE EIGENVECTORS
C OF THE HESSENBERG MATRIX ARE DESIRED, SET ORTR(J) AND
C ORTI(J) TO 0.0D0 FOR THESE ELEMENTS.
C
C HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX.
C THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN FURTHER
C INFORMATION ABOUT THE TRANSFORMATIONS WHICH WERE USED IN THE
C REDUCTION BY CORTH, IF PERFORMED. IF THE EIGENVECTORS OF
C THE HESSENBERG MATRIX ARE DESIRED, THESE ELEMENTS MAY BE
C ARBITRARY.
C
C ON OUTPUT
C
C ORTR, ORTI, AND THE UPPER HESSENBERG PORTIONS OF HR AND HI
C HAVE BEEN DESTROYED.
C
C WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE EIGENVALUES. IF AN ERROR
C EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT
C FOR INDICES IERR+1,...,N.
C
C ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE EIGENVECTORS. THE EIGENVECTORS
C ARE UNNORMALIZED. IF AN ERROR EXIT IS MADE, NONE OF
C THE EIGENVECTORS HAS BEEN FOUND.
C
C IERR IS SET TO
C ZERO FOR NORMAL RETURN,
C J IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED
C WHILE THE J-TH EIGENVALUE IS BEING SOUGHT.
C
C CALLS CDIV FOR COMPLEX DIVISION.
C CALLS CSROOT FOR COMPLEX SQUARE ROOT.
C CALLS PYTHAG FOR DSQRT(A*A + B*B) .
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
c
c unnecessary initialization of L to keep g77 -Wall happy
c
L = 0
C
IERR = 0
C .......... INITIALIZE EIGENVECTOR MATRIX ..........
DO 101 J = 1, N
C
DO 100 I = 1, N
ZR(I,J) = 0.0D0
ZI(I,J) = 0.0D0
100 CONTINUE
ZR(J,J) = 1.0D0
101 CONTINUE
C .......... FORM THE MATRIX OF ACCUMULATED TRANSFORMATIONS
C FROM THE INFORMATION LEFT BY CORTH ..........
IEND = IGH - LOW - 1
IF (IEND) 180, 150, 105
C .......... FOR I=IGH-1 STEP -1 UNTIL LOW+1 DO -- ..........
105 DO 140 II = 1, IEND
I = IGH - II
IF (ORTR(I) .EQ. 0.0D0 .AND. ORTI(I) .EQ. 0.0D0) GO TO 140
IF (HR(I,I-1) .EQ. 0.0D0 .AND. HI(I,I-1) .EQ. 0.0D0) GO TO 140
C .......... NORM BELOW IS NEGATIVE OF H FORMED IN CORTH ..........
NORM = HR(I,I-1) * ORTR(I) + HI(I,I-1) * ORTI(I)
IP1 = I + 1
C
DO 110 K = IP1, IGH
ORTR(K) = HR(K,I-1)
ORTI(K) = HI(K,I-1)
110 CONTINUE
C
DO 130 J = I, IGH
SR = 0.0D0
SI = 0.0D0
C
DO 115 K = I, IGH
SR = SR + ORTR(K) * ZR(K,J) + ORTI(K) * ZI(K,J)
SI = SI + ORTR(K) * ZI(K,J) - ORTI(K) * ZR(K,J)
115 CONTINUE
C
SR = SR / NORM
SI = SI / NORM
C
DO 120 K = I, IGH
ZR(K,J) = ZR(K,J) + SR * ORTR(K) - SI * ORTI(K)
ZI(K,J) = ZI(K,J) + SR * ORTI(K) + SI * ORTR(K)
120 CONTINUE
C
130 CONTINUE
C
140 CONTINUE
C .......... CREATE REAL SUBDIAGONAL ELEMENTS ..........
150 L = LOW + 1
C
DO 170 I = L, IGH
LL = MIN0(I+1,IGH)
IF (HI(I,I-1) .EQ. 0.0D0) GO TO 170
NORM = PYTHAG(HR(I,I-1),HI(I,I-1))
YR = HR(I,I-1) / NORM
YI = HI(I,I-1) / NORM
HR(I,I-1) = NORM
HI(I,I-1) = 0.0D0
C
DO 155 J = I, N
SI = YR * HI(I,J) - YI * HR(I,J)
HR(I,J) = YR * HR(I,J) + YI * HI(I,J)
HI(I,J) = SI
155 CONTINUE
C
DO 160 J = 1, LL
SI = YR * HI(J,I) + YI * HR(J,I)
HR(J,I) = YR * HR(J,I) - YI * HI(J,I)
HI(J,I) = SI
160 CONTINUE
C
DO 165 J = LOW, IGH
SI = YR * ZI(J,I) + YI * ZR(J,I)
ZR(J,I) = YR * ZR(J,I) - YI * ZI(J,I)
ZI(J,I) = SI
165 CONTINUE
C
170 CONTINUE
C .......... STORE ROOTS ISOLATED BY CBAL ..........
180 DO 200 I = 1, N
IF (I .GE. LOW .AND. I .LE. IGH) GO TO 200
WR(I) = HR(I,I)
WI(I) = HI(I,I)
200 CONTINUE
C
EN = IGH
TR = 0.0D0
TI = 0.0D0
ITN = 30*N
C .......... SEARCH FOR NEXT EIGENVALUE ..........
220 IF (EN .LT. LOW) GO TO 680
ITS = 0
ENM1 = EN - 1
C .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT
C FOR L=EN STEP -1 UNTIL LOW DO -- ..........
240 DO 260 LL = LOW, EN
L = EN + LOW - LL
IF (L .EQ. LOW) GO TO 300
TST1 = DABS(HR(L-1,L-1)) + DABS(HI(L-1,L-1))
X + DABS(HR(L,L)) + DABS(HI(L,L))
TST2 = TST1 + DABS(HR(L,L-1))
IF (TST2 .EQ. TST1) GO TO 300
260 CONTINUE
C .......... FORM SHIFT ..........
300 IF (L .EQ. EN) GO TO 660
IF (ITN .EQ. 0) GO TO 1000
IF (ITS .EQ. 10 .OR. ITS .EQ. 20) GO TO 320
SR = HR(EN,EN)
SI = HI(EN,EN)
XR = HR(ENM1,EN) * HR(EN,ENM1)
XI = HI(ENM1,EN) * HR(EN,ENM1)
IF (XR .EQ. 0.0D0 .AND. XI .EQ. 0.0D0) GO TO 340
YR = (HR(ENM1,ENM1) - SR) / 2.0D0
YI = (HI(ENM1,ENM1) - SI) / 2.0D0
CALL CSROOT(YR**2-YI**2+XR,2.0D0*YR*YI+XI,ZZR,ZZI)
IF (YR * ZZR + YI * ZZI .GE. 0.0D0) GO TO 310
ZZR = -ZZR
ZZI = -ZZI
310 CALL CDIV(XR,XI,YR+ZZR,YI+ZZI,XR,XI)
SR = SR - XR
SI = SI - XI
GO TO 340
C .......... FORM EXCEPTIONAL SHIFT ..........
320 SR = DABS(HR(EN,ENM1)) + DABS(HR(ENM1,EN-2))
SI = 0.0D0
C
340 DO 360 I = LOW, EN
HR(I,I) = HR(I,I) - SR
HI(I,I) = HI(I,I) - SI
360 CONTINUE
C
TR = TR + SR
TI = TI + SI
ITS = ITS + 1
ITN = ITN - 1
C .......... REDUCE TO TRIANGLE (ROWS) ..........
LP1 = L + 1
C
DO 500 I = LP1, EN
SR = HR(I,I-1)
HR(I,I-1) = 0.0D0
NORM = PYTHAG(PYTHAG(HR(I-1,I-1),HI(I-1,I-1)),SR)
XR = HR(I-1,I-1) / NORM
WR(I-1) = XR
XI = HI(I-1,I-1) / NORM
WI(I-1) = XI
HR(I-1,I-1) = NORM
HI(I-1,I-1) = 0.0D0
HI(I,I-1) = SR / NORM
C
DO 490 J = I, N
YR = HR(I-1,J)
YI = HI(I-1,J)
ZZR = HR(I,J)
ZZI = HI(I,J)
HR(I-1,J) = XR * YR + XI * YI + HI(I,I-1) * ZZR
HI(I-1,J) = XR * YI - XI * YR + HI(I,I-1) * ZZI
HR(I,J) = XR * ZZR - XI * ZZI - HI(I,I-1) * YR
HI(I,J) = XR * ZZI + XI * ZZR - HI(I,I-1) * YI
490 CONTINUE
C
500 CONTINUE
C
SI = HI(EN,EN)
IF (SI .EQ. 0.0D0) GO TO 540
NORM = PYTHAG(HR(EN,EN),SI)
SR = HR(EN,EN) / NORM
SI = SI / NORM
HR(EN,EN) = NORM