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*DECK ALPLSF
SUBROUTINE ALPLSF(IMETH,N,X,Y,EPSRID,LREAL,NOR,A,B,PREC)
*
*-----------------------------------------------------------------------
*
*Purpose:
* compute the polynomial coefficients of a Pade approximation using a
* direct least square procedure.
*
*Copyright:
* Copyright (C) 1996 Ecole Polytechnique de Montreal
* This library is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version
*
*Author(s): A. Hebert
*
*Parameters: input
* IMETH type of algorithm (=1: use QR; =2: use SVD; =3: use NNLS).
* N number of collocation points.
* X abscissa of the collocation points.
* Y ordinates of the collocation points.
* EPSRID epsilon used in polynomial simplification.
* LREAL selection flag (=.true. to get rid of complex roots).
*
*Parameters: output
* NOR order of the polynomials.
* A polynomial coefficients of the numerator of the Pade
* approximation. A(0) is the constant term.
* B polynomial coefficients of the denominator of the Pade
* approximation. B(0) is the constant term.
* DOUBLE PRECISION A(0:NOR),B(0:NOR)
* PREC accuracy of the fit.
*
*-----------------------------------------------------------------------
*
*----
* SUBROUTINE ARGUMENTS
*----
INTEGER IMETH,N,NOR
REAL X(N),Y(N),PREC
DOUBLE PRECISION EPSRID,A(0:(N-1)/2),B(0:(N-1)/2)
LOGICAL LREAL
*----
* LOCAL VARIABLES
*----
PARAMETER (MAXNOR=10,MAXPTS=99)
DOUBLE PRECISION BB(MAXPTS),GAR,PARAM(MAXPTS),AGAR(0:MAXNOR),
1 BGAR(0:MAXNOR),CGAR(0:MAXNOR+1),W(MAXPTS),RV1(MAXPTS),SGN,
2 GAROLD,YAPPR,RNORM,RMAX
COMPLEX*16 SIGX0(MAXNOR+1),SIGXW(MAXNOR+1),DDAGAR(0:MAXNOR),
1 DDBGAR(0:MAXNOR),WEIGH(MAXNOR+1),CC,DD,CCC,XCC,DCC
LOGICAL LFAIL
DOUBLE PRECISION, ALLOCATABLE, DIMENSION(:,:) :: AA,V
*----
* SCRATCH STORAGE ALLOCATION
*----
ALLOCATE(AA(MAXPTS,MAXPTS),V(MAXPTS,MAXPTS))
*
IF(N.GT.MAXPTS) CALL XABORT('ALPLSF: INSUFFICIENT MAXPTS.')
*
* NOR=0 CASE.
GAROLD=0.0
NOR=0
PREC=0.0
A(0)=DBLE(Y(N))
B(0)=1.0D0
DO 10 I=1,N-1
PREC=MAX(PREC,ABS(Y(N)/Y(I)-1.0))
10 CONTINUE
*
RMAX=1.0D10
DO 210 IINOR=1,MIN((N-1)/2,MAXNOR)
INOR=IINOR
IF(X(N).GE.0.99E10) THEN
DO 41 I=1,N-1
GAR=1.0D0
IOF=0
DO 20 J=0,INOR-1
IOF=IOF+1
AA(I,IOF)=GAR
GAR=GAR*X(I)
20 CONTINUE
GAR=1.0D0
DO 30 J=0,INOR-1
IOF=IOF+1
GAROLD=GAR
AA(I,IOF)=-GAR*Y(I)
GAR=GAR*X(I)
30 CONTINUE
BB(I)=(Y(I)-Y(N))*X(I)
DO 40 J=1,2*INOR
AA(I,J)=AA(I,J)/GAROLD
40 CONTINUE
41 CONTINUE
IF(IMETH.EQ.1) THEN
CALL ALST2F(MAXPTS,N-1,2*INOR,AA,RV1)
CALL ALST2S(MAXPTS,N-1,2*INOR,AA,RV1,BB,PARAM)
ELSE IF(IMETH.EQ.2) THEN
CALL ALSVDF(AA,N-1,2*INOR,MAXPTS,MAXPTS,W,V,RV1)
DO 45 J=1,2*INOR
IF(W(J).EQ.0.0D0) CALL XABORT('ALPLSF: SVD FAILURE(1).')
45 CONTINUE
CALL ALSVDS(AA,W,V,N-1,2*INOR,MAXPTS,MAXPTS,BB,PARAM,RV1)
ELSE IF(IMETH.EQ.3) THEN
CALL ALNNLS(AA,N-1,2*INOR,MAXPTS,MAXPTS,BB,PARAM,RNORM,MODE)
IF(MODE.NE.1) CALL XABORT('ALPLSF: NNLS FAILURE(1).')
IF((INOR.GT.1).AND.(RNORM.GE.0.95D0*RMAX)) GO TO 210
RMAX=RNORM
ENDIF
DO 50 I=0,INOR-1
AGAR(I)=PARAM(I+1)
BGAR(I)=PARAM(INOR+1+I)
50 CONTINUE
AGAR(INOR)=Y(N)
BGAR(INOR)=1.0D0
ELSE
DO 81 I=1,N
GAR=1.0D0
IOF=0
DO 60 J=0,INOR
IOF=IOF+1
AA(I,IOF)=GAR
GAR=GAR*X(I)
60 CONTINUE
GAR=1.0D0
DO 70 J=0,INOR-1
IOF=IOF+1
GAROLD=GAR
AA(I,IOF)=-GAR*Y(I)
GAR=GAR*X(I)
70 CONTINUE
BB(I)=Y(I)*X(I)
DO 80 J=1,2*INOR+1
AA(I,J)=AA(I,J)/GAROLD
80 CONTINUE
81 CONTINUE
IF(IMETH.EQ.1) THEN
CALL ALST2F(MAXPTS,N,2*INOR+1,AA,RV1)
CALL ALST2S(MAXPTS,N,2*INOR+1,AA,RV1,BB,PARAM)
ELSE IF(IMETH.EQ.2) THEN
CALL ALSVDF(AA,N,2*INOR+1,MAXPTS,MAXPTS,W,V,RV1)
DO 85 J=1,2*INOR
IF(W(J).EQ.0.0D0) CALL XABORT('ALPLSF: SVD FAILURE(2).')
85 CONTINUE
CALL ALSVDS(AA,W,V,N,2*INOR+1,MAXPTS,MAXPTS,BB,PARAM,RV1)
ELSE IF(IMETH.EQ.3) THEN
CALL ALNNLS(AA,N,2*INOR+1,MAXPTS,MAXPTS,BB,PARAM,RNORM,MODE)
IF(MODE.NE.1) CALL XABORT('ALPLSF: NNLS FAILURE(2).')
IF((INOR.GT.1).AND.(RNORM.GE.0.95D0*RMAX)) GO TO 210
RMAX=RNORM
ENDIF
DO 90 I=0,INOR
AGAR(I)=PARAM(I+1)
IF(I.EQ.INOR) THEN
BGAR(I)=1.0D0
ELSE
BGAR(I)=PARAM(INOR+2+I)
ENDIF
90 CONTINUE
ENDIF
*
* POLYNOMIAL SIMPLIFICATION.
DDAGAR(0)=AGAR(INOR)
DDBGAR(0)=BGAR(INOR)
CALL ALROOT(AGAR(0:INOR),INOR,SIGX0,LFAIL)
IF(LFAIL) GO TO 210
CALL ALROOT(BGAR(0:INOR),INOR,SIGXW,LFAIL)
IF(LFAIL) GO TO 210
IJINOR=1
95 XXX=REAL(ABS(DBLE(SIGXW(IJINOR))-DBLE(SIGX0(IJINOR))))
IF(XXX.LT.EPSRID*ABS(DBLE(SIGXW(IJINOR)))) THEN
INOR=INOR-1
DO 100 I=IJINOR,INOR
SIGX0(I)=SIGX0(I+1)
SIGXW(I)=SIGXW(I+1)
100 CONTINUE
ELSE IF((DBLE(SIGXW(IJINOR)).GT.EPSRID).AND.
> (DIMAG(SIGXW(IJINOR)).EQ.0.0).AND.
> (IMETH.EQ.3)) THEN
CALL XABORT('ALPLSF: NNLS FAILURE(3).')
ELSE IF((DBLE(SIGXW(IJINOR)).GT.0.1*EPSRID).AND.
> (DIMAG(SIGXW(IJINOR)).EQ.0.0)) THEN
GO TO 210
ELSE
IJINOR=IJINOR+1
ENDIF
IF(IJINOR.LE.INOR) GO TO 95
IF(INOR.LT.0) CALL XABORT('ALPLSF: ALGORITHM FAILURE.')
DO 120 I=1,INOR
DDAGAR(I)=DDAGAR(I-1)
DDBGAR(I)=DDBGAR(I-1)
DO 110 J=I-1,1,-1
DDAGAR(J)=DDAGAR(J-1)-DDAGAR(J)*SIGX0(I)
DDBGAR(J)=DDBGAR(J-1)-DDBGAR(J)*SIGXW(I)
110 CONTINUE
DDAGAR(0)=-DDAGAR(0)*SIGX0(I)
DDBGAR(0)=-DDBGAR(0)*SIGXW(I)
120 CONTINUE
DO 130 I=0,INOR
AGAR(I)=DBLE(DDAGAR(I))/DBLE(DDBGAR(INOR))
BGAR(I)=DBLE(DDBGAR(I))/DBLE(DDBGAR(INOR))
IF(AGAR(I).LE.0.0D0) GO TO 210
IF(BGAR(I).LE.0.0D0) GO TO 210
130 CONTINUE
SGN=1.0D0
CGAR(0)=AGAR(0)
DO 135 I=2,INOR+1
SGN=-SGN
CGAR(I-1)=SGN*(BGAR(I-2)+AGAR(I-1))
135 CONTINUE
CGAR(INOR+1)=-SGN
CALL ALROOT(CGAR(0:INOR+1),INOR+1,SIGX0,LFAIL)
IF(LFAIL) GO TO 210
*
* NEWTON IMPROVEMENT OF THE ROOTS.
DO 138 I=1,INOR+1
CCC=0.0D0
XCC=1.0D0
DO 136 J=0,INOR+1
CCC=CCC+CGAR(J)*XCC
XCC=XCC*SIGX0(I)
136 CONTINUE
DCC=0.0D0
XCC=1.0D0
DO 137 J=1,INOR+1
DCC=DCC+CGAR(J)*XCC*REAL(J)
XCC=XCC*SIGX0(I)
137 CONTINUE
SIGX0(I)=SIGX0(I)-CCC/DCC
138 CONTINUE
*
IF(LREAL) THEN
DO 140 I=1,INOR+1
IF(DBLE(SIGX0(I)).LT.1.0E-10) GO TO 210
IF(DIMAG(SIGX0(I)).NE.0.0) GO TO 210
140 CONTINUE
ENDIF
*
* COMPUTE THE WEIGHTS.
DO 170 I=1,INOR+1
CC=(1.0D0,0.0D0)
DD=0.0D0
DO 150 JNOR=0,INOR
DD=DD+BGAR(JNOR)*CC
CC=-CC*SIGX0(I)
150 CONTINUE
DO 160 J=1,INOR+1
IF(J.NE.I) DD=DD/(SIGX0(J)-SIGX0(I))
160 CONTINUE
WEIGH(I)=DD
170 CONTINUE
*
* TEST THE ACCURACY OF THE PADE APPROXIMATION.
PREC1=0.0
DO 190 I=1,N
CC=0.0D0
DD=0.0D0
DO 180 JNOR=1,INOR+1
CC=CC+WEIGH(JNOR)/(SIGX0(JNOR)+X(I))
DD=DD+WEIGH(JNOR)*SIGX0(JNOR)/(SIGX0(JNOR)+X(I))
180 CONTINUE
YAPPR=DBLE(DD/CC)
PREC1=MAX(PREC1,ABS(REAL(YAPPR)/Y(I)-1.0))
190 CONTINUE
*
IF(PREC1.LT.0.95*PREC) THEN
NOR=INOR
PREC=PREC1
DO 200 I=0,NOR
A(I)=AGAR(I)
B(I)=BGAR(I)
200 CONTINUE
ENDIF
210 CONTINUE
*----
* SCRATCH STORAGE DEALLOCATION
*----
DEALLOCATE(V,AA)
RETURN
END
|
