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SUBROUTINE sla_FITXY (ITYPE,NP,XYE,XYM,COEFFS,J)
*+
* - - - - - -
* F I T X Y
* - - - - - -
*
* Fit a linear model to relate two sets of [X,Y] coordinates.
*
* Given:
* ITYPE i type of model: 4 or 6 (note 1)
* NP i number of samples (note 2)
* XYE d(2,np) expected [X,Y] for each sample
* XYM d(2,np) measured [X,Y] for each sample
*
* Returned:
* COEFFS d(6) coefficients of model (note 3)
* J i status: 0 = OK
* -1 = illegal ITYPE
* -2 = insufficient data
* -3 = singular solution
*
* Notes:
*
* 1) ITYPE, which must be either 4 or 6, selects the type of model
* fitted. Both allowed ITYPE values produce a model COEFFS which
* consists of six coefficients, namely the zero points and, for
* each of XE and YE, the coefficient of XM and YM. For ITYPE=6,
* all six coefficients are independent, modelling squash and shear
* as well as origin, scale, and orientation. However, ITYPE=4
* selects the "solid body rotation" option; the model COEFFS
* still consists of the same six coefficients, but now two of
* them are used twice (appropriately signed). Origin, scale
* and orientation are still modelled, but not squash or shear -
* the units of X and Y have to be the same.
*
* 2) For NC=4, NP must be at least 2. For NC=6, NP must be at
* least 3.
*
* 3) The model is returned in the array COEFFS. Naming the
* elements of COEFFS as follows:
*
* COEFFS(1) = A
* COEFFS(2) = B
* COEFFS(3) = C
* COEFFS(4) = D
* COEFFS(5) = E
* COEFFS(6) = F
*
* the model is:
*
* XE = A + B*XM + C*YM
* YE = D + E*XM + F*YM
*
* For the "solid body rotation" option (ITYPE=4), the
* magnitudes of B and F, and of C and E, are equal. The
* signs of these coefficients depend on whether there is a
* sign reversal between XE,YE and XM,YM; fits are performed
* with and without a sign reversal and the best one chosen.
*
* 4) Error status values J=-1 and -2 leave COEFFS unchanged;
* if J=-3 COEFFS may have been changed.
*
* See also sla_PXY, sla_INVF, sla_XY2XY, sla_DCMPF
*
* Called: sla_DMAT, sla_DMXV
*
* P.T.Wallace Starlink 11 February 1991
*
* Copyright (C) 1995 Rutherford Appleton Laboratory
*-
IMPLICIT NONE
INTEGER ITYPE,NP
DOUBLE PRECISION XYE(2,NP),XYM(2,NP),COEFFS(6)
INTEGER J
INTEGER I,JSTAT,IW(4),NSOL
DOUBLE PRECISION P,SXE,SXEXM,SXEYM,SYE,SYEYM,SYEXM,SXM,
: SYM,SXMXM,SXMYM,SYMYM,XE,YE,
: XM,YM,V(4),DM3(3,3),DM4(4,4),DET,
: SGN,SXXYY,SXYYX,SX2Y2,A,B,C,D,
: SDR2,XR,YR,AOLD,BOLD,COLD,DOLD,SOLD
* Preset the status
J=0
* Float the number of samples
P=DBLE(NP)
* Check ITYPE
IF (ITYPE.EQ.6) THEN
*
* Six-coefficient linear model
* ----------------------------
* Check enough samples
IF (NP.GE.3) THEN
* Form summations
SXE=0D0
SXEXM=0D0
SXEYM=0D0
SYE=0D0
SYEYM=0D0
SYEXM=0D0
SXM=0D0
SYM=0D0
SXMXM=0D0
SXMYM=0D0
SYMYM=0D0
DO I=1,NP
XE=XYE(1,I)
YE=XYE(2,I)
XM=XYM(1,I)
YM=XYM(2,I)
SXE=SXE+XE
SXEXM=SXEXM+XE*XM
SXEYM=SXEYM+XE*YM
SYE=SYE+YE
SYEYM=SYEYM+YE*YM
SYEXM=SYEXM+YE*XM
SXM=SXM+XM
SYM=SYM+YM
SXMXM=SXMXM+XM*XM
SXMYM=SXMYM+XM*YM
SYMYM=SYMYM+YM*YM
END DO
* Solve for A,B,C in XE = A + B*XM + C*YM
V(1)=SXE
V(2)=SXEXM
V(3)=SXEYM
DM3(1,1)=P
DM3(1,2)=SXM
DM3(1,3)=SYM
DM3(2,1)=SXM
DM3(2,2)=SXMXM
DM3(2,3)=SXMYM
DM3(3,1)=SYM
DM3(3,2)=SXMYM
DM3(3,3)=SYMYM
CALL sla_DMAT(3,DM3,V,DET,JSTAT,IW)
IF (JSTAT.EQ.0) THEN
DO I=1,3
COEFFS(I)=V(I)
END DO
* Solve for D,E,F in YE = D + E*XM + F*YM
V(1)=SYE
V(2)=SYEXM
V(3)=SYEYM
CALL sla_DMXV(DM3,V,COEFFS(4))
ELSE
* No 6-coefficient solution possible
J=-3
END IF
ELSE
* Insufficient data for 6-coefficient fit
J=-2
END IF
ELSE IF (ITYPE.EQ.4) THEN
*
* Four-coefficient solid body rotation model
* ------------------------------------------
* Check enough samples
IF (NP.GE.2) THEN
* Try two solutions, first without then with flip in X
DO NSOL=1,2
IF (NSOL.EQ.1) THEN
SGN=1D0
ELSE
SGN=-1D0
END IF
* Form summations
SXE=0D0
SXXYY=0D0
SXYYX=0D0
SYE=0D0
SXM=0D0
SYM=0D0
SX2Y2=0D0
DO I=1,NP
XE=XYE(1,I)*SGN
YE=XYE(2,I)
XM=XYM(1,I)
YM=XYM(2,I)
SXE=SXE+XE
SXXYY=SXXYY+XE*XM+YE*YM
SXYYX=SXYYX+XE*YM-YE*XM
SYE=SYE+YE
SXM=SXM+XM
SYM=SYM+YM
SX2Y2=SX2Y2+XM*XM+YM*YM
END DO
*
* Solve for A,B,C,D in: +/- XE = A + B*XM - C*YM
* + YE = D + C*XM + B*YM
V(1)=SXE
V(2)=SXXYY
V(3)=SXYYX
V(4)=SYE
DM4(1,1)=P
DM4(1,2)=SXM
DM4(1,3)=-SYM
DM4(1,4)=0D0
DM4(2,1)=SXM
DM4(2,2)=SX2Y2
DM4(2,3)=0D0
DM4(2,4)=SYM
DM4(3,1)=SYM
DM4(3,2)=0D0
DM4(3,3)=-SX2Y2
DM4(3,4)=-SXM
DM4(4,1)=0D0
DM4(4,2)=SYM
DM4(4,3)=SXM
DM4(4,4)=P
CALL sla_DMAT(4,DM4,V,DET,JSTAT,IW)
IF (JSTAT.EQ.0) THEN
A=V(1)
B=V(2)
C=V(3)
D=V(4)
* Determine sum of radial errors squared
SDR2=0D0
DO I=1,NP
XM=XYM(1,I)
YM=XYM(2,I)
XR=A+B*XM-C*YM-XYE(1,I)*SGN
YR=D+C*XM+B*YM-XYE(2,I)
SDR2=SDR2+XR*XR+YR*YR
END DO
ELSE
* Singular: set flag
SDR2=-1D0
END IF
* If first pass and non-singular, save variables
IF (NSOL.EQ.1.AND.JSTAT.EQ.0) THEN
AOLD=A
BOLD=B
COLD=C
DOLD=D
SOLD=SDR2
END IF
END DO
* Pick the best of the two solutions
IF (SOLD.GE.0D0.AND.(SOLD.LE.SDR2.OR.NP.EQ.2)) THEN
COEFFS(1)=AOLD
COEFFS(2)=BOLD
COEFFS(3)=-COLD
COEFFS(4)=DOLD
COEFFS(5)=COLD
COEFFS(6)=BOLD
ELSE IF (JSTAT.EQ.0) THEN
COEFFS(1)=-A
COEFFS(2)=-B
COEFFS(3)=C
COEFFS(4)=D
COEFFS(5)=C
COEFFS(6)=B
ELSE
* No 4-coefficient fit possible
J=-3
END IF
ELSE
* Insufficient data for 4-coefficient fit
J=-2
END IF
ELSE
* Illegal ITYPE - not 4 or 6
J=-1
END IF
END
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