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ORTHREGF.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME ORTHREGF
* Problem :
* *********
* An orthogonal regression problem
* The problem is to fit (orthogonally) an torus to a
* set of points in 3D space. This set of points is generated by
* perturbing a first set lying exactly on a predefined torus
* centered at the origin.
* Source:
* M. Gulliksson,
* "Algorithms for nonlinear Least-squares with Applications to
* Orthogonal Regression",
* UMINF-178.90, University of Umea, Sweden, 1990.
* SIF input: Ph. Toint, June 1990.
* minor correction by Ph. Shott, Jan 1995.
* classification QOR2-AY-V-V
* square root of the number of data points
* (number of variables = 3 * NPTS**2 + 5 )
*IE NPTS 5 $-PARAMETER n = 80 original value
*IE NPTS 7 $-PARAMETER n = 152
*IE NPTS 10 $-PARAMETER n = 305
*IE NPTS 15 $-PARAMETER n = 680
*IE NPTS 20 $-PARAMETER n = 1205
IE NPTS 40 $-PARAMETER n = 4805
* True torus parameters (centered at the origin)
RE TP4 1.7
RE TP5 0.8
* Perturbation parameters
RE PSEED 237.1531
RE PSIZE 0.2
* Constants
IE 1 1
IE 5 5
RE PI 3.1415926535
* Computed parameters
RM 2PI PI 2.0
RI RNPTS NPTS
RD ICR0 RNPTS 1.0
R* INCR ICR0 2PI
* Construct the data points
DO I 1 NPTS
IA I-1 I -1
RI RI-1 I-1
R* THETA1 RI-1 INCR
R( ST1 SIN THETA1
R( CT1 COS THETA1
R* P5CT1 TP5 CT1
R+ P4P5CT1 TP4 P5CT1
R* R3 TP5 ST1
DO J 1 NPTS
IA J-1 J -1
RI RJ-1 J-1
R* THETA2 RJ-1 INCR
R( ST2 SIN THETA2
R( CT2 COS THETA2
R* R1 P4P5CT1 CT2
R* R2 P4P5CT1 ST2
R* XSEED THETA2 PSEED
R( SSEED COS XSEED
R* PER-1 PSIZE SSEED
RA PERT PER-1 1.0
A* XD(I,J) R1 PERT
A* YD(I,J) R2 PERT
A* ZD(I,J) R3 PERT
ND
VARIABLES
* Parameters of the torus
DO I 1 5
X P(I)
ND
* Projections of the data points onto the torus
DO I 1 NPTS
DO J 1 NPTS
X X(I,J)
X Y(I,J)
X Z(I,J)
ND
GROUPS
N OBJ
DO I 1 NPTS
DO J 1 NPTS
XN OX(I,J) X(I,J) 1.0
XN OY(I,J) Y(I,J) 1.0
XN OZ(I,J) Z(I,J) 1.0
XE A(I,J)
ND
CONSTANTS
DO I 1 NPTS
DO J 1 NPTS
Z ORTHREGF OX(I,J) XD(I,J)
Z ORTHREGF OY(I,J) YD(I,J)
Z ORTHREGF OZ(I,J) ZD(I,J)
ND
BOUNDS
FR ORTHREGF 'DEFAULT'
XL ORTHREGF P4 0.001
XL ORTHREGF P5 0.001
START POINT
ORTHREGF P1 1.0
ORTHREGF P2 0.0
ORTHREGF P3 1.0
ORTHREGF P4 1.0
ORTHREGF P5 0.5
DO I 1 NPTS
DO J 1 NPTS
Z ORTHREGF X(I,J) XD(I,J)
Z ORTHREGF Y(I,J) YD(I,J)
Z ORTHREGF Z(I,J) ZD(I,J)
ND
ELEMENT TYPE
EV TA XX YY
EV TA A B
EV TA C
IV TA XMA YMB
IV TA CC
EV ISQ Z P
IV ISQ ZMP
EV SQ XX
ELEMENT USES
DO I 1 NPTS
DO J 1 NPTS
XT EA(I,J) TA
ZV EA(I,J) XX X(I,J)
ZV EA(I,J) YY Y(I,J)
ZV EA(I,J) A P1
ZV EA(I,J) B P2
ZV EA(I,J) C P4
XT EB(I,J) ISQ
ZV EB(I,J) Z Z(I,J)
ZV EB(I,J) P P3
XT EC(I,J) SQ
ZV EC(I,J) XX P5
ND
GROUP TYPE
GV L2 GVAR
GROUP USES
DO I 1 NPTS
DO J 1 NPTS
XT OX(I,J) L2
XT OY(I,J) L2
XT OZ(I,J) L2
XE A(I,J) EA(I,J) EB(I,J)
XE A(I,J) EC(I,J) -1.0
ND
OBJECT BOUND
LO ORTHREGF 0.0
* Solution
*LO SOLTN(5) 0.990089426
*LO SOLTN(7) 1.315031322
*LO SOLTN(10) 4.515848902
*LO SOLTN(15) 9.185538338
*LO SOLTN(20) 16.20054380
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS ORTHREGF
TEMPORARIES
R CCSQ
R CCCB
R XXYY
R T
R DTDX
R DTDY
R DTDC
R D2TDX2
R D2TDY2
R D2TDC2
R D2TDXC
R D2TDYC
R S
R DSDX
R DSDY
R DSDC
R D2SDX2
R D2SDY2
R D2SDC2
R D2SDXY
R D2SDXC
R D2SDYC
R R
R SS
R SPS
M SQRT
INDIVIDUALS
T TA
R XMA XX 1.0 A -1.0
R YMB YY 1.0 B -1.0
R CC C 1.0
A CCSQ CC * CC
A CCCB CCSQ * CC
A XXYY XMA * XMA + YMB * YMB
A T XXYY / CCSQ
A DTDX 2.0 * XMA / CCSQ
A DTDY 2.0 * YMB / CCSQ
A DTDC - 2.0 * XXYY / CCCB
A D2TDX2 2.0 / CCSQ
A D2TDY2 2.0 / CCSQ
A D2TDC2 6.0 * XXYY / ( CCSQ * CCSQ)
A D2TDXC - 4.0 * XMA / CCCB
A D2TDYC - 4.0 * YMB / CCCB
A S SQRT( T )
A R 0.5 / S
A DSDX R * DTDX
A DSDY R * DTDY
A DSDC R * DTDC
A D2SDX2 R * ( D2TDX2 - 0.5 * DTDX * DTDX / T )
A D2SDY2 R * ( D2TDY2 - 0.5 * DTDY * DTDY / T )
A D2SDC2 R * ( D2TDC2 - 0.5 * DTDC * DTDC / T )
A D2SDXY -0.5 * DTDX * DSDY / T
A D2SDXC R * ( D2TDXC - 0.5 * DTDX * DTDC / T )
A D2SDYC R * ( D2TDYC - 0.5 * DTDY * DTDC / T )
A SS S - 1.0
A SPS SS + SS
F SS * SS
G XMA SPS * DSDX
G YMB SPS * DSDY
G CC SPS * DSDC
H XMA XMA SPS * D2SDX2 + 2.0 * DSDX * DSDX
H XMA YMB SPS * D2SDXY + 2.0 * DSDX * DSDY
H XMA CC SPS * D2SDXC + 2.0 * DSDX * DSDC
H YMB YMB SPS * D2SDY2 + 2.0 * DSDY * DSDY
H YMB CC SPS * D2SDYC + 2.0 * DSDY * DSDC
H CC CC SPS * D2SDC2 + 2.0 * DSDC * DSDC
T ISQ
R ZMP Z 1.0 P -1.0
F ZMP * ZMP
G ZMP ZMP + ZMP
H ZMP ZMP 2.0
T SQ
F XX * XX
G XX XX + XX
H XX XX 2.0
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS ORTHREGF
* Least-square groups
INDIVIDUALS
T L2
F GVAR * GVAR
G GVAR + GVAR
H 2.0
ENDATA