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egrad_brh2.f
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egrad_brh2.f
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
! CARACAL - Ring polymer molecular dynamics and rate constant calculations
! on black-box generated potential energy surfaces
!
! Copyright (c) 2023 by Julien Steffen ([email protected])
! Stefan Grimme ([email protected]) (QMDFF code)
!
! Permission is hereby granted, free of charge, to any person obtaining a
! copy of this software and associated documentation files (the "Software"),
! to deal in the Software without restriction, including without limitation
! the rights to use, copy, modify, merge, publish, distribute, sublicense,
! and/or sell copies of the Software, and to permit persons to whom the
! Software is furnished to do so, subject to the following conditions:
!
! The above copyright notice and this permission notice shall be included in
! all copies or substantial portions of the Software.
!
! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
! IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
! THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
! FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
! DEALINGS IN THE SOFTWARE.
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Potential energy routine taken from the POTLIB library:
! R. J. Duchovic, Y. L. Volobuev, G. C. Lynch, A. W. Jasper, D. G. Truhlar, T. C. Allison,
! A. F. Wagner, B. C. Garrett, J. Espinosa-García, and J. C. Corchado, POTLIB,
! https://comp.chem.umn.edu/potlib.
subroutine egrad_brh2(q,Natoms,Nbeads,V,dVdq,info)
integer, intent(in) :: Natoms
integer, intent(in) :: Nbeads
double precision, intent(in) :: q(3,Natoms,Nbeads)
double precision, intent(out) :: V(Nbeads)
double precision, intent(out) :: dVdq(3,Natoms,Nbeads)
double precision :: R(3), dVdr(3)
double precision :: xAB, yAB, zAB, rAB
double precision :: xAC, yAC, zAC, rAC
double precision :: xBC, yBC, zBC, rBC
integer k, info
info = 0
do k = 1, Nbeads
xAB = q(1,2,k) - q(1,1,k)
yAB = q(2,2,k) - q(2,1,k)
zAB = q(3,2,k) - q(3,1,k)
rAB = sqrt(xAB * xAB + yAB * yAB + zAB * zAB)
R(1) = rAB
xAC = q(1,1,k) - q(1,3,k)
yAC = q(2,1,k) - q(2,3,k)
zAC = q(3,1,k) - q(3,3,k)
rAC = sqrt(xAC * xAC + yAC * yAC + zAC * zAC)
R(2) = rAC
xBC = q(1,3,k) - q(1,2,k)
yBC = q(2,3,k) - q(2,2,k)
zBC = q(3,3,k) - q(3,2,k)
rBC = sqrt(xBC * xBC + yBC * yBC + zBC * zBC)
R(3) = rBC
call pot_brh2(R, V(k), dVdr)
dVdq(1,1,k) = dVdr(2) * xAC / rAC - dVdr(1) * xAB / rAB
dVdq(2,1,k) = dVdr(2) * yAC / rAC - dVdr(1) * yAB / rAB
dVdq(3,1,k) = dVdr(2) * zAC / rAC - dVdr(1) * zAB / rAB
dVdq(1,2,k) = dVdr(1) * xAB / rAB - dVdr(3) * xBC / rBC
dVdq(2,2,k) = dVdr(1) * yAB / rAB - dVdr(3) * yBC / rBC
dVdq(3,2,k) = dVdr(1) * zAB / rAB - dVdr(3) * zBC / rBC
dVdq(1,3,k) = dVdr(3) * xBC / rBC - dVdr(2) * xAC / rAC
dVdq(2,3,k) = dVdr(3) * yBC / rBC - dVdr(2) * yAC / rAC
dVdq(3,3,k) = dVdr(3) * zBC / rBC - dVdr(2) * zAC / rAC
end do
c stop "in pooott"
end subroutine egrad_brh2
SUBROUTINE initialize_brh2
C
C System: BrH2
C Functional form: Diatomics-in-molecules plus three-center term
C Common name: BrH2 DIM-3C
C Interface: 3-1S
C References: D. C. Clary
C Chem. Phys. 71, 117 (1982)
C I. Last and M. Baer
C in Potential Energy Surfaces and Dynamics
C edited by D. G. Truhlar (Plenum, New York, 1981)
C p. 519
C
C Number of bodies: 3
C Number of derivatives: 1
C Number of electronic surfaces: 1
C
C Calling Sequence:
C PREPOT - initializes the potential's variables and
C must be called once before any calls to POT
C POT - driver for the evaluation of the energy and the derivatives
C of the energy with respect to the coordinates for a given
C geometric configuration
C
C Units:
C energies - hartrees
C coordinates - bohr
C derivatives - hartrees/bohr
C
C Surfaces:
C ground electronic state
C
C Zero of energy:
C The classical potential energy is set equal to zero for the Br
C infinitely far from the H2 diatomic and R(H2) set equal to the
C H2 equilibrium diatomic value.
C
C Parameters:
C Set in data statements
C
C Coordinates:
C Internal, Definition: R(1) = R(Br-H)
C R(2) = R(H-H)
C R(3) = R(Br-H)
C
C Common Blocks (used between the calling program and this potential):
C /PT1CM/ R(3), ENERGY, DEDR(3)
C passes the coordinates, ground state electronic energy, and
C derivatives of the ground electronic state energy with respect
C to the coordinates.
C /PT4CM/ IPTPRT, IDUM(19)
C passes the FORTRAN unit number used for output from the potential
C /PT5CM/ EASYAB, EASYBC, EASYAC
C passes the energy in the three asymptotic valleys for an A + BC system.
C The energy in the AB valley, EASYAB, is equal to the energy of the
C C atom "infinitely" far from the AB diatomic and R(AB) set equal to
C Re(AB), the equilibrium bond length for the AB diatomic.
C In this potential the AB valley represents H infinitely far from
C the BrH diatomic and R(BrH) equal to Re(BrH). Similarly, the terms
C EASYBC and EASYAC represent the energies in the H2 and the HBr
C valleys, respectively.
C
C Default Parameter Values:
C Variable Default value
C IPTPRT 6
C
C*****
C
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
c COMMON /PT1CM/ R(3), ENERGY, DEDR(3)
COMMON /PT4CM_brh2/ IPTPRT, IDUM(19)
COMMON /PT5CM_brh2/ EASYAB, EASYBC, EASYAC
C
C
LOGICAL LCOL
DIMENSION H(10),DH1(10),DH2(10),DH3(10),U(4,4),E(4),SCR1(4),
1 SCR2(4),R(3),dedr(3)
COMMON /POTCOM/ EPS,ONE,RHH,AHH,DHH,BHH,XIH,XIX,G,ALFW,RHX,AHX,
& DHX,BHX,ETAHH,ETA3S,ETA1P,ETA3P
save
C
C Echo potential parameters to FORTRAN unit IPTPRT
c WRITE (IPTPRT, 600) RHH,AHH,DHH,BHH,RHX,AHX,DHX,BHX
c WRITE (IPTPRT, 610) ETAHH,ETA3S,ETA1P,ETA3P,
c * XIH,XIX,G,ALFW
C
EASYAB = DHX
EASYBC = DHH
EASYAC = DHX
C
XIB = 0.5D0*(XIH + XIX)
T = XIB*RHX
C3 = 1.D0/3.D0
DENOM = 2.D0*RHX*(1.D0+T*(1.D0+C3*T))*EXP(-T)
G = G/RHX
RT34 = 0.25D0*SQRT(3.D0)
RT38 = 0.5D0*RT34
C
600 FORMAT(/,1X,'*****','Potential Energy Surface',1X,'*****',
* //,1X,T5,'BrH2 DIM-3C potential energy surface',
* //,1X,T5,'Potential energy surface parameters:',
* /,1X,T5,'HH parameters (atomic units):',
* /,1X,T10,'RHH',T16,'=',T18,1PE13.5,
* T40,'AHH',T46,'=',T48,1PE13.5,
* /,1X,T10,'DHH',T16,'=',T18,1PE13.5,
* T40,'BHH',T46,'=',T48,1PE13.5,
* /,1X,T5,'HBr parameters (atomic units):',
* /,1X,T10,'RHBr',T16,'=',T18,1PE13.5,
* T40,'AHBr',T46,'=',T48,1PE13.5,
* /,1X,T10,'DHBr',T16,'=',T18,1PE13.5,
* T40,'BHBr',T46,'=',T48,1PE13.5)
610 FORMAT(/,1X,T5,'Other parameters (atomic units):',
* /,1X,T10,'ETAHH',T16,'=',T18,1PE13.5,
* T40,'ETA3S',T46,'=',T48,1PE13.5,
* /,1X,T10,'ETA1P',T16,'=',T18,1PE13.5,
* T40,'ETA3P',T46,'=',T48,1PE13.5,
* /,1X,T10,'XIH',T16,'=',T18,1PE13.5,T40,'XIX',T46,'=',T48,1PE13.5,
* /,1X,T10,'G',T16,'=',T18,1PE13.5,T40,'ALFW',T46,'=',T48,1PE13.5,
* //,1X,'*****')
C
RETURN
c end
C Changed to new subroutine layout with dummy parameters for calling
entry POT_brh2(R,energy,dedr)
c IMPLICIT DOUBLE PRECISION (A-H,O-Z)
c real(kind=8)::R(3),dedr(3),energy
c COMMON /PT4CM/ IPTPRT, IDUM(19)
c COMMON /PT5CM/ EASYAB, EASYBC, EASYAC
C
C The potential routines define R1=R(Br-H), R2=R(H-Br), and R3=R(H-H);
C rearrange the input coordinates to match these definitions
C
R1 = R(1)
R2 = R(3)
R3 = R(2)
C
IF(R1.GT.R2+R3) R1 = R2+R3
IF(R2.GT.R1+R3) R2 = R1+R3
IF(R3.GT.R1+R2) R3 = R1+R2
R1S = R1*R1
R2S = R2*R2
R3S = R3*R3
R12 = R1*R2
R13 = R1*R3
R23 = R2*R3
T3 = (R1S+R2S-R3S)*0.5D0
T2 = (R1S+R3S-R2S)*0.5D0
T1 = (R2S+R3S-R1S)*0.5D0
CSA = T2/R13
IF(ABS(CSA) .GT. ONE) CSA = SIGN(ONE,CSA)
CSA2 = CSA*CSA
SNA2 = 1.D0-CSA2
LCOL = SNA2 .LT. EPS
SNA = SQRT(SNA2)
T22 = 2.D0*CSA
T11 = 0.D0
IF(.NOT.LCOL) T11 = 2.D0*(SNA2-CSA2)/SNA
SN2A = T22*SNA
T = T3/(R1*R13)
DCSA21 = T22*T
DSN2A1 = T11*T
T = -R2/R13
DCSA22 = T22*T
DSN2A2 = T11*T
T = T1/(R13*R3)
DCSA23 = T22*T
DSN2A3 = T11*T
CSB = T1/R23
IF(ABS(CSB).GT.ONE) CSB = SIGN(ONE,CSB)
CSB2 = CSB*CSB
SNB2 = 1.D0-CSB2
SNB = SQRT(SNB2)
T11 = 0.D0
IF(.NOT.LCOL) T11 = 2.D0*(SNB2-CSB2)/SNB
T22 = 2.D0*CSB
SN2B = T22*SNB
T = -R1/R23
DCSB21 = T22*T
DSN2B1 = T11*T
T = T3/(R2*R23)
DCSB22 = T22*T
DSN2B2 = T11*T
T = T2/(R23*R3)
DCSB23 = T22*T
DSN2B3 = T11*T
T = T3/R12
CSG2 = T*T
T = 2.D0*T
DCSG21 = T*T2/(R1*R12)
DCSG22 = T*T1/(R12*R2)
DCSG23 = -T*R3/R12
C
C DIATOMIC CURVES
C HH
RDIF = R3 - RHH
EX1 = EXP(-AHH*RDIF)
RDIF2 = RDIF*RDIF
EX2 = EXP(-BHH*RDIF*RDIF2)
T1 = DHH*EX1*EX2
V1HH = T1*(EX1-2.D0)
V3HH = ETAHH*T1*(EX1+2.D0)
T1 = 2.D0*AHH*T1
T2 = 3.D0*BHH*RDIF2
DV1HH = T1*(1.D0-EX1) - T2*V1HH
DV3HH = -T1*ETAHH*(1.D0+EX1) - T2*V3HH
C HX
CALL VHX(R1,V1S1,V3S1,V1P1,V3P1,DV1S1,DV3S1,DV1P1,DV3P1)
CALL VHX(R2,V1S2,V3S2,V1P2,V3P2,DV1S2,DV3S2,DV1P2,DV3P2)
C
S11 = V1S1 + 3.D0*V3S1
S12 = V1S2 + 3.D0*V3S2
S21 = 3.D0*V1S1 + V3S1
S22 = 3.D0*V1S2 + V3S2
S31 = V1S1 - V3S1
S32 = V1S2 - V3S2
P11 = V1P1 + 3.D0*V3P1
P12 = V1P2 + 3.D0*V3P2
P21 = 3.D0*V1P1 + V3P1
P22 = 3.D0*V1P2 + V3P2
P31 = V1P1 - V3P1
P32 = V1P2 - V3P2
C
DS11 = DV1S1 + 3.D0*DV3S1
DS12 = DV1S2 + 3.D0*DV3S2
DS21 = 3.D0*DV1S1 + DV3S1
DS22 = 3.D0*DV1S2 + DV3S2
DS31 = DV1S1 - DV3S1
DS32 = DV1S2 - DV3S2
DP11 = DV1P1 + 3.D0*DV3P1
DP12 = DV1P2 + 3.D0*DV3P2
DP21 = 3.D0*DV1P1 + DV3P1
DP22 = 3.D0*DV1P2 + DV3P2
DP31 = DV1P1 - DV3P1
DP32 = DV1P2 - DV3P2
C
C CONSTRUCT 4X4 HAMILTONIAN PUT IN PACKED ARRAY
C
H(1) = V1HH + 0.25D0*(S11*CSA2 + P11*SNA2 + S12*CSB2 + P12*SNB2)
H(3) = V1HH + 0.25D0*(S11*SNA2 + P11*CSA2 + S12*SNB2 + P12*CSB2)
H(6) = V3HH + 0.25D0*(S21*CSA2 + P21*SNA2 + S22*CSB2 + P22*SNB2)
H(10) = V3HH + 0.25D0*(S21*SNA2 + P21*CSA2 + S22*SNB2 + P22*CSB2)
T11 = S11 - P11
T12 = S12 - P12
H(2) = 0.D0
IF(.NOT.LCOL) H(2) = 0.125D0*(T11*SN2A - T12*SN2B)
H(4) = RT34*(S31*CSA2 + P31*SNA2 - S32*CSB2 - P32*SNB2)
T31 = S31 - P31
T32 = S32 - P32
H(5) = 0.D0
IF(.NOT.LCOL) H(5) = RT38*(T31*SN2A + T32*SN2B)
H(7) = H(5)
H(8) = RT34*(S31*SNA2 + P31*CSA2 - S32*SNB2 - P32*CSB2)
T21 = S21 - P21
T22 = S22 - P22
H(9) = 0.D0
IF(.NOT.LCOL) H(9) = 0.125D0*(T21*SN2A - T22*SN2B)
C
C CONSTRUCT DERIVATIVE OF HAMILTONIAN MATRIX
T = T11*DCSA21 + T12*DCSB21
DH1(1) = 0.25D0*(DS11*CSA2 + DP11*SNA2 + T)
DH1(2) = 0.125D0*((DS11-DP11)*SN2A + T11*DSN2A1 - T12*DSN2B1)
DH1(3) = 0.25D0*(DS11*SNA2 + DP11*CSA2 - T)
T = T11*DCSA22 + T12*DCSB22
DH2(1) = 0.25D0*(DS12*CSB2 + DP12*SNB2 + T)
DH2(2) = 0.125D0*(-(DS12-DP12)*SN2B + T11*DSN2A2 - T12*DSN2B2)
DH2(3) = 0.25D0*(DS12*SNB2 + DP12*CSB2 - T)
T = 0.25D0*(T11*DCSA23 + T12*DCSB23)
DH3(1) = DV1HH + T
DH3(2) = 0.125D0*(T11*DSN2A3 - T12*DSN2B3)
DH3(3) = DV1HH - T
T = T21*DCSA21 + T22*DCSB21
DH1(6) = 0.25D0*(DS21*CSA2 + DP21*SNA2 + T)
DH1(9) = 0.125D0*((DS21-DP21)*SN2A + T21*DSN2A1 - T22*DSN2B1)
DH1(10) = 0.25D0*(DS21*SNA2 + DP21*CSA2 - T)
T = T21*DCSA22 + T22*DCSB22
DH2(6) = 0.25D0*(DS22*CSB2 + DP22*SNB2 + T)
DH2(9) = 0.125D0*((DP22-DS22)*SN2B + T21*DSN2A2 - T22*DSN2B2)
DH2(10) = 0.25D0*(DS22*SNB2 + DP22*CSB2 - T)
T = 0.25D0*(T21*DCSA23 + T22*DCSB23)
DH3(6) = DV3HH + T
DH3(9) = 0.125D0*(T21*DSN2A3 - T22*DSN2B3)
DH3(10) = DV3HH - T
T = T31*DCSA21 - T32*DCSB21
DH1(4) = RT34*(DS31*CSA2 + DP31*SNA2 + T)
DH1(5) = RT38*((DS31-DP31)*SN2A + T31*DSN2A1 + T32*DSN2B1)
DH1(7) = DH1(5)
DH1(8) = RT34*(DS31*SNA2 + DP31*CSA2 - T)
T = T31*DCSA22 - T32*DCSB22
DH2(4) = RT34*(-DS32*CSB2 - DP32*SNB2 + T)
DH2(5) = RT38*((DS32-DP32)*SN2B + T31*DSN2A2 + T32*DSN2B2)
DH2(7) = DH2(5)
DH2(8) = RT34*(-DS32*SNB2 - DP32*CSB2 - T)
T = T31*DCSA23 - T32*DCSB23
DH3(4) = RT34*T
DH3(5) = RT38*(T31*DSN2A3 + T32*DSN2B3)
DH3(7) = DH3(5)
DH3(8) = -DH3(4)
C
C DIAGONALIZE H PUT EIGENVECTORS INTO U, EIGENVALUES INTO E
C USE EISPACK ROUTINE RSP
C
CALL RSP(4,4,10,H,E,1,U,SCR1,SCR2,IERR)
IF(IERR.NE.0) THEN
WRITE(IPTPRT,6000) R1,R2,R3,H
WRITE(IPTPRT,6001) V1HH,V3HH,V1S1,V3S1,V1S2,
* V3S2,V1P1,V3P1,V1P2,V3P2
C
STOP 'POT 3'
END IF
C EVALUATE DERIVATIVES OF LOWEST ROOT
D1 = 0.D0
D2 = 0.D0
D3 = 0.D0
DO 20 L = 1,4
T1 = U(L,1)
LL = (L*(L-1))/2
DO 25 K = 1,4
T2 = T1*U(K,1)
KK = (K*(K-1))/2
IF(L.GE.K) INDEX = LL + K
IF(K.GT.L) INDEX = KK + L
D1 = D1 + T2*DH1(INDEX)
D2 = D2 + T2*DH2(INDEX)
D3 = D3 + T2*DH3(INDEX)
25 CONTINUE
20 CONTINUE
C
C SUPPLEMENTARY TERM
T = XIH*R3
EX1 = EXP(-T)
SHH = (1.D0+T*(1.D0+C3*T))*EX1
DSHH = -XIH*T*(1.D0+T)*C3*EX1
T = XIB*R1
EX1 = EXP(-T)
SHX1 = R1*(1.D0+T*(1.D0+C3*T))*EX1/DENOM
DSHX1 = (1.D0+T*(1.D0-C3*T*T))*EX1/DENOM
T = XIB*R2
EX1 = EXP(-T)
SHX2 = R2*(1.D0+T*(1.D0+C3*T))*EX1/DENOM
DSHX2 = (1.D0+T*(1.D0-C3*T*T))*EX1/DENOM
RDIF = R1 - R2
T1 = G*EXP(-ALFW*RDIF*RDIF)
T2 = SHH*(SHX1+SHX2) + SHX1*SHX2
ENERGY = E(1) + T2*T1*CSG2 + DHH
T3 = 2.D0*ALFW*RDIF*CSG2
DEDR(1) = D1 + (DSHX1*(SHH+SHX2)*CSG2 + T2*(DCSG21-T3))*T1
DEDR(3) = D2 + (DSHX2*(SHH+SHX1)*CSG2 + T2*(DCSG22+T3))*T1
DEDR(2) = D3 + (DSHH*(SHX1+SHX2)*CSG2 + T2*DCSG23)*T1
C
6000 FORMAT(/,2X,'Error returned by the subprogram RSP in POT',
* /,2X,'R = ',T10,3(1PE13.5,1X),
* /,2X,'H = ',(T10,4(1PE13.5,1X)))
6001 FORMAT(2X,'V1HH,V3HH,V1S1,V3S1,V1S2,V3S2,V1P1,V3P1,V1P2,V3P2=',
* /,2X,(T10,4(1PE13.5,1X)))
C
RETURN
END
SUBROUTINE VHX(R,V1S,V3S,V1P,V3P,DV1S,DV3S,DV1P,DV3P)
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
COMMON /POTCOM/ EPS,ONE,RHH,AHH,DHH,BHH,XIH,XIX,G,ALFW,RHX,AHX,
& DHX,BHX,ETAHH,ETA3S,ETA1P,ETA3P
RDIF = R - RHX
RDIF2 = RDIF*RDIF
EX1 = EXP(-AHX*RDIF)
EX2 = EXP(-BHX*RDIF2*RDIF)
T1 = DHX*EX1*EX2
V1S = T1*(EX1-2.D0)
V1 = T1*(EX1+2.D0)
V3S = ETA3S*V1
V1P = ETA1P*V1
V3P = ETA3P*V1
T1 = 2.D0*AHX*T1
T2 = 3.D0*BHX*RDIF2
DV1S = T1*(1.D0-EX1) - T2*V1S
DV1 = -T1*(1.D0+EX1) - T2*V1
DV3S = ETA3S*DV1
DV1P = ETA1P*DV1
DV3P = ETA3P*DV1
RETURN
END
C
BLOCK DATA PTPARM
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
COMMON /PT4CM_brh2/ IPTPRT, IDUM(19)
COMMON /POTCOM/ EPS,ONE,RHH,AHH,DHH,BHH,XIH,XIX,G,ALFW,RHX,AHX,
& DHX,BHX,ETAHH,ETA3S,ETA1P,ETA3P
DATA EPS /1.D-6/, ONE /1.0D0/
DATA RHH,AHH,DHH,BHH /1.4016D0, 1.0291D0, 0.17447D0, 0.018D0/
DATA XIH,XIX,G,ALFW /1.D0, 1.6D0, 0.22D0, 1.0D0/
DATA IPTPRT /6/
DATA RHX /2.673D0/
DATA AHX /0.957D0/
DATA DHX /0.1439D0/
DATA BHX /0.012D0/
DATA ETAHH /0.393764D0/
DATA ETA3S /0.322D0/
DATA ETA1P /0.20286D0/
DATA ETA3P /0.1771D0/
END
C
SUBROUTINE RSP(NM,N,NV,A,W,MATZ,Z,FV1,FV2,IERR)
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DOUBLE PRECISION A(NV),W(N),Z(NM,N),FV1(N),FV2(N)
C
C THIS SUBROUTINE CALLS THE RECOMMENDED SEQUENCE OF
C SUBROUTINES FROM THE EIGENSYSTEM SUBROUTINE PACKAGE (EISPACK)
C TO FIND THE EIGENVALUES AND EIGENVECTORS (IF DESIRED)
C OF A REAL SYMMETRIC PACKED MATRIX.
C
C ON INPUT-
C
C NM MUST BE SET TO THE ROW DIMENSION OF THE TWO-DIMENSIONAL
C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C DIMENSION STATEMENT,
C
C N IS THE ORDER OF THE MATRIX A,
C
C NV IS AN INTEGER VARIABLE SET EQUAL TO THE
C DIMENSION OF THE ARRAY A AS SPECIFIED FOR
C A IN THE CALLING PROGRAM. NV MUST NOT BE
C LESS THAN N*(N+1)/2,
C
C A CONTAINS THE LOWER TRIANGLE OF THE REAL SYMMETRIC
C PACKED MATRIX STORED ROW-WISE,
C
C MATZ IS AN INTEGER VARIABLE SET EQUAL TO ZERO IF
C ONLY EIGENVALUES ARE DESIRED, OTHERWISE IT IS SET TO
C ANY NON-ZERO INTEGER FOR BOTH EIGENVALUES AND EIGENVECTORS.
C
C ON OUTPUT-
C
C W CONTAINS THE EIGENVALUES IN ASCENDING ORDER,
C
C Z CONTAINS THE EIGENVECTORS IF MATZ IS NOT ZERO,
C
C IERR IS AN INTEGER OUTPUT VARIABLE SET EQUAL TO AN
C ERROR COMPLETION CODE DESCRIBED IN SECTION 2B OF THE
C DOCUMENTATION. THE NORMAL COMPLETION CODE IS ZERO,
C
C FV1 AND FV2 ARE TEMPORARY STORAGE ARRAYS.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO B. S. GARBOW,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C
C ------------------------------------------------------------------
C
DATA ZERO,ONE/0.0D0,1.0D0/
IF (N .LE. NM) GO TO 5
IERR = 10 * N
GO TO 50
5 IF (NV .GE. (N * (N + 1)) / 2) GO TO 10
IERR = 20 * N
GO TO 50
C
10 CALL TRED3(N,NV,A,W,FV1,FV2)
IF (MATZ .NE. 0) GO TO 20
C ********** FIND EIGENVALUES ONLY **********
CALL TQLRAT_br(N,W,FV2,IERR)
GO TO 50
C ********** FIND BOTH EIGENVALUES AND EIGENVECTORS **********
20 DO 40 I = 1, N
C
DO 30 J = 1, N
Z(J,I) = ZERO
30 CONTINUE
C
Z(I,I) = ONE
40 CONTINUE
C
CALL TQL2_br(NM,N,W,FV1,Z,IERR)
IF (IERR .NE. 0) GO TO 50
CALL TRBAK3(NM,N,NV,A,N,Z)
50 RETURN
C ********** LAST CARD OF RSP **********
END
SUBROUTINE TRED3(N,NV,A,D,E,E2)
C
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DOUBLE PRECISION A(NV),D(N),E(N),E2(N)
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRED3,
C NUM. MATH. 11, 181-195(1968) BY MARTIN, REINSCH, AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C THIS SUBROUTINE REDUCES A REAL SYMMETRIC MATRIX, STORED AS
C A ONE-DIMENSIONAL ARRAY, TO A SYMMETRIC TRIDIAGONAL MATRIX
C USING ORTHOGONAL SIMILARITY TRANSFORMATIONS.
C
C ON INPUT-
C
C N IS THE ORDER OF THE MATRIX,
C
C NV MUST BE SET TO THE DIMENSION OF THE ARRAY PARAMETER A
C AS DECLARED IN THE CALLING PROGRAM DIMENSION STATEMENT,
C
C A CONTAINS THE LOWER TRIANGLE OF THE REAL SYMMETRIC
C INPUT MATRIX, STORED ROW-WISE AS A ONE-DIMENSIONAL
C ARRAY, IN ITS FIRST N*(N+1)/2 POSITIONS.
C
C ON OUTPUT-
C
C A CONTAINS INFORMATION ABOUT THE ORTHOGONAL
C TRANSFORMATIONS USED IN THE REDUCTION,
C
C D CONTAINS THE DIAGONAL ELEMENTS OF THE TRIDIAGONAL MATRIX,
C
C E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE TRIDIAGONAL
C MATRIX IN ITS LAST N-1 POSITIONS. E(1) IS SET TO ZERO,
C
C E2 CONTAINS THE SQUARES OF THE CORRESPONDING ELEMENTS OF E.
C E2 MAY COINCIDE WITH E IF THE SQUARES ARE NOT NEEDED.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO B. S. GARBOW,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C
C ------------------------------------------------------------------
C
C ********** FOR I=N STEP -1 UNTIL 1 DO -- **********
DATA ZERO/0.0D0/
DO 300 II = 1, N
I = N + 1 - II
L = I - 1
IZ = (I * L) / 2
H = ZERO
SCALE = ZERO
IF (L .LT. 1) GO TO 130
C ********** SCALE ROW (ALGOL TOL THEN NOT NEEDED) **********
DO 120 K = 1, L
IZ = IZ + 1
D(K) = A(IZ)
SCALE = SCALE + ABS(D(K))
120 CONTINUE
C
IF (SCALE .NE. ZERO) GO TO 140
130 E(I) = ZERO
E2(I) = ZERO
GO TO 290
C
140 DO 150 K = 1, L
D(K) = D(K) / SCALE
H = H + D(K) * D(K)
150 CONTINUE
C
E2(I) = SCALE * SCALE * H
F = D(L)
G = -SIGN(SQRT(H),F)
E(I) = SCALE * G
H = H - F * G
D(L) = F - G
A(IZ) = SCALE * D(L)
IF (L .EQ. 1) GO TO 290
F = ZERO
C
DO 240 J = 1, L
G = ZERO
JK = (J * (J-1)) / 2
C ********** FORM ELEMENT OF A*U **********
DO 180 K = 1, L
JK = JK + 1
IF (K .GT. J) JK = JK + K - 2
G = G + A(JK) * D(K)
180 CONTINUE
C ********** FORM ELEMENT OF P **********
E(J) = G / H
F = F + E(J) * D(J)
240 CONTINUE
C
HH = F / (H + H)
JK = 0
C ********** FORM REDUCED A **********
DO 260 J = 1, L
F = D(J)
G = E(J) - HH * F
E(J) = G
C
DO 265 K = 1, J
JK = JK + 1
A(JK) = A(JK) - F * E(K) - G * D(K)
265 CONTINUE
260 CONTINUE
C
290 D(I) = A(IZ+1)
A(IZ+1) = SCALE * SQRT(H)
300 CONTINUE
C
RETURN
C ********** LAST CARD OF TRED3 **********
END
SUBROUTINE TQLRAT_br(N,D,E2,IERR)
C
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION D(N),E2(N)
DOUBLE PRECISION MACHEP
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TQLRAT_br,
C ALGORITHM 464, COMM. ACM 16, 689(1973) BY REINSCH.
C
C THIS SUBROUTINE FINDS THE EIGENVALUES OF A SYMMETRIC
C TRIDIAGONAL MATRIX BY THE RATIONAL QL METHOD.
C
C ON INPUT-
C
C N IS THE ORDER OF THE MATRIX,
C
C D CONTAINS THE DIAGONAL ELEMENTS OF THE INPUT MATRIX,
C
C E2 CONTAINS THE SQUARES OF THE SUBDIAGONAL ELEMENTS OF THE
C INPUT MATRIX IN ITS LAST N-1 POSITIONS. E2(1) IS ARBITRARY.
C
C ON OUTPUT-
C
C D CONTAINS THE EIGENVALUES IN ASCENDING ORDER. IF AN
C ERROR EXIT IS MADE, THE EIGENVALUES ARE CORRECT AND
C ORDERED FOR INDICES 1,2,...IERR-1, BUT MAY NOT BE
C THE SMALLEST EIGENVALUES,
C
C E2 HAS BEEN DESTROYED,
C
C IERR IS SET TO
C ZERO FOR NORMAL RETURN,
C J IF THE J-TH EIGENVALUE HAS NOT BEEN
C DETERMINED AFTER 30 ITERATIONS.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO B. S. GARBOW,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C
C ------------------------------------------------------------------
C
C **********
DATA ZERO,ONE,TWO/0.0D0,1.0D0,2.0D0/
C ********** MACHEP IS A MACHINE DEPENDENT PARAMETER SPECIFYING
C THE RELATIVE PRECISION OF FLOATING POINT ARITHMETIC.
C
MACHEP = TWO**(-37)
C
IERR = 0
IF (N .EQ. 1) GO TO 1001
C
DO 100 I = 2, N
E2(I-1) = E2(I)
100 CONTINUE
C
F = ZERO
B = ZERO
E2(N) = ZERO
C
DO 290 L = 1, N
J = 0
H = MACHEP * (ABS(D(L)) + SQRT(E2(L)))
IF (B .GT. H) GO TO 105
B = H
C = B * B
C ********** LOOK FOR SMALL SQUARED SUB-DIAGONAL ELEMENT **********
105 DO 110 M = L, N
IF (E2(M) .LE. C) GO TO 120
C ********** E2(N) IS ALWAYS ZERO, SO THERE IS NO EXIT
C THROUGH THE BOTTOM OF THE LOOP **********
110 CONTINUE
C
120 IF (M .EQ. L) GO TO 210
130 IF (J .EQ. 30) GO TO 1000
J = J + 1
C ********** FORM SHIFT **********
L1 = L + 1
S = SQRT(E2(L))
G = D(L)
P = (D(L1) - G) / (TWO * S)
R = SQRT(P*P+ONE)
D(L) = S / (P + SIGN(R,P))
H = G - D(L)
C
DO 140 I = L1, N
D(I) = D(I) - H
140 CONTINUE
C
F = F + H
C ********** RATIONAL QL TRANSFORMATION **********
G = D(M)
IF (G .EQ. ZERO) G = B
H = G
S = ZERO
MML = M - L
C ********** FOR I=M-1 STEP -1 UNTIL L DO -- **********
DO 200 II = 1, MML
I = M - II
P = G * H
R = P + E2(I)
E2(I+1) = S * R
S = E2(I) / R
D(I+1) = H + S * (H + D(I))
G = D(I) - E2(I) / G
IF (G .EQ. ZERO) G = B
H = G * P / R
200 CONTINUE
C
E2(L) = S * G
D(L) = H
C ********** GUARD AGAINST UNDERFLOW IN CONVERGENCE TEST **********
IF (H .EQ. ZERO) GO TO 210
IF (ABS(E2(L)) .LE. ABS(C/H)) GO TO 210
E2(L) = H * E2(L)
IF (E2(L) .NE. ZERO) GO TO 130
210 P = D(L) + F
C ********** ORDER EIGENVALUES **********
IF (L .EQ. 1) GO TO 250
C ********** FOR I=L STEP -1 UNTIL 2 DO -- **********
DO 230 II = 2, L
I = L + 2 - II
IF (P .GE. D(I-1)) GO TO 270
D(I) = D(I-1)
230 CONTINUE
C
250 I = 1
270 D(I) = P
290 CONTINUE
C
GO TO 1001
C ********** SET ERROR -- NO CONVERGENCE TO AN
C EIGENVALUE AFTER 30 ITERATIONS **********
1000 IERR = L
1001 RETURN
C ********** LAST CARD OF TQLRAT_br **********
END
SUBROUTINE TQL2_br(NM,N,D,E,Z,IERR)
C
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION D(N),E(N),Z(NM,N)
DOUBLE PRECISION MACHEP
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TQL2_br,
C NUM. MATH. 11, 293-306(1968) BY BOWDLER, MARTIN, REINSCH, AND
C WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971).
C
C THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS
C OF A SYMMETRIC TRIDIAGONAL MATRIX BY THE QL METHOD.
C THE EIGENVECTORS OF A FULL SYMMETRIC MATRIX CAN ALSO
C BE FOUND IF TRED2 HAS BEEN USED TO REDUCE THIS
C FULL MATRIX TO TRIDIAGONAL 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 D CONTAINS THE DIAGONAL ELEMENTS OF THE INPUT MATRIX,
C
C E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE INPUT MATRIX
C IN ITS LAST N-1 POSITIONS. E(1) IS ARBITRARY,
C
C Z CONTAINS THE TRANSFORMATION MATRIX PRODUCED IN THE
C REDUCTION BY TRED2, IF PERFORMED. IF THE EIGENVECTORS
C OF THE TRIDIAGONAL MATRIX ARE DESIRED, Z MUST CONTAIN
C THE IDENTITY MATRIX.
C
C ON OUTPUT-
C
C D CONTAINS THE EIGENVALUES IN ASCENDING ORDER. IF AN
C ERROR EXIT IS MADE, THE EIGENVALUES ARE CORRECT BUT
C UNORDERED FOR INDICES 1,2,...,IERR-1,
C
C E HAS BEEN DESTROYED,
C
C Z CONTAINS ORTHONORMAL EIGENVECTORS OF THE SYMMETRIC
C TRIDIAGONAL (OR FULL) MATRIX. IF AN ERROR EXIT IS MADE,
C Z CONTAINS THE EIGENVECTORS ASSOCIATED WITH THE STORED
C EIGENVALUES,
C
C IERR IS SET TO
C ZERO FOR NORMAL RETURN,
C J IF THE J-TH EIGENVALUE HAS NOT BEEN
C DETERMINED AFTER 30 ITERATIONS.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO B. S. GARBOW,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C
C ------------------------------------------------------------------
C
C **********
DATA ZERO,ONE,TWO/0.0D0,1.0D0,2.0D0/
C ********** MACHEP IS A MACHINE DEPENDENT PARAMETER SPECIFYING
C THE RELATIVE PRECISION OF FLOATING POINT ARITHMETIC.
C
MACHEP = TWO**(-37)
C
IERR = 0
IF (N .EQ. 1) GO TO 1001
C
DO 100 I = 2, N
E(I-1) = E(I)
100 CONTINUE
C
F = ZERO
B = ZERO
E(N) = ZERO
C
DO 240 L = 1, N
J = 0
H = MACHEP * (ABS(D(L)) + ABS(E(L)))
IF (B .LT. H) B = H
C ********** LOOK FOR SMALL SUB-DIAGONAL ELEMENT **********
DO 110 M = L, N
IF (ABS(E(M)) .LE. B) GO TO 120
C ********** E(N) IS ALWAYS ZERO, SO THERE IS NO EXIT
C THROUGH THE BOTTOM OF THE LOOP **********
110 CONTINUE
C
120 IF (M .EQ. L) GO TO 220
130 IF (J .EQ. 30) GO TO 1000
J = J + 1
C ********** FORM SHIFT **********
L1 = L + 1
G = D(L)
P = (D(L1) - G) / (TWO * E(L))
R = SQRT(P*P+ONE)
D(L) = E(L) / (P + SIGN(R,P))
H = G - D(L)
C
DO 140 I = L1, N
D(I) = D(I) - H
140 CONTINUE
C
F = F + H
C ********** QL TRANSFORMATION **********
P = D(M)
C = ONE
S = ZERO
MML = M - L
C ********** FOR I=M-1 STEP -1 UNTIL L DO -- **********
DO 200 II = 1, MML
I = M - II
G = C * E(I)
H = C * P
IF (ABS(P) .LT. ABS(E(I))) GO TO 150
C = E(I) / P
R = SQRT(C*C+ONE)
E(I+1) = S * P * R
S = C / R
C = ONE / R
GO TO 160
150 C = P / E(I)
R = SQRT(C*C+ONE)
E(I+1) = S * E(I) * R
S = ONE / R
C = C * S
160 P = C * D(I) - S * G
D(I+1) = H + S * (C * G + S * D(I))
C ********** FORM VECTOR **********
DO 180 K = 1, N
H = Z(K,I+1)
Z(K,I+1) = S * Z(K,I) + C * H
Z(K,I) = C * Z(K,I) - S * H
180 CONTINUE
C
200 CONTINUE
C
E(L) = S * P
D(L) = C * P
IF (ABS(E(L)) .GT. B) GO TO 130
220 D(L) = D(L) + F
240 CONTINUE
C ********** ORDER EIGENVALUES AND EIGENVECTORS **********
DO 300 II = 2, N
I = II - 1
K = I
P = D(I)
C
DO 260 J = II, N
IF (D(J) .GE. P) GO TO 260
K = J
P = D(J)
260 CONTINUE
C
IF (K .EQ. I) GO TO 300
D(K) = D(I)
D(I) = P
C
DO 280 J = 1, N
P = Z(J,I)
Z(J,I) = Z(J,K)
Z(J,K) = P
280 CONTINUE
C
300 CONTINUE
C
GO TO 1001
C ********** SET ERROR -- NO CONVERGENCE TO AN
C EIGENVALUE AFTER 30 ITERATIONS **********
1000 IERR = L
1001 RETURN
C ********** LAST CARD OF TQL2_br **********
END
SUBROUTINE TRBAK3(NM,N,NV,A,M,Z)
C
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION A(NV),Z(NM,M)
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRBAK3,
C NUM. MATH. 11, 181-195(1968) BY MARTIN, REINSCH, AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C THIS SUBROUTINE FORMS THE EIGENVECTORS OF A REAL SYMMETRIC
C MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
C SYMMETRIC TRIDIAGONAL MATRIX DETERMINED BY TRED3.
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