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ORTHREGC.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME ORTHREGC
* Problem :
* *********
* An orthogonal regression problem,
* The problem is to fit (orthogonally) an ellipse to a set of points
* in the plane. This set of points is generated by perturbing a
* first set lying exactly on a predefined ellipse centered at the
* origin.
* Source: adapted from:
* 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.
* classification QQR2-AN-V-V
* Number of data points
* (number of variables = 2 NPTS + 5 )
*IE NPTS 10 $-PARAMETER n= 25 original value
*IE NPTS 50 $-PARAMETER n= 105
*IE NPTS 250 $-PARAMETER n= 505
*IE NPTS 500 $-PARAMETER n= 1005
IE NPTS 2500 $-PARAMETER n= 5005
*IE NPTS 5000 $-PARAMETER n= 10005
* True ellipse parameters (centered at the origin)
RE V11 2.0
RE V12 1.0
RE V22 2.0
* Perturbation parameters
RE PSEED 237.1531
RE PSIZE 0.2
* Constants
IE 1 1
IE 0 0
RE PI 3.1415926535
* Computed parameters
RM 2PI PI 2.0
RI RNPTS NPTS
RD ICR0 RNPTS 1.0
R* INCR ICR0 2PI
R( C3 COS V22
R( S3 SIN V22
RM -S3 S3 -1.0
* Construct the data points
DO I 1 NPTS
IA I-1 I -1
RI RI-1 I-1
R* THETA RI-1 INCR
R( ST SIN THETA
R( CT COS THETA
R* U1 V11 CT
R* U2 V12 ST
R* U1C U1 C3
R* U2-S U2 -S3
R+ R1 U1C U2-S
R* U1S U1 S3
R* U2C U2 C3
R+ R2 U1S U2C
R* XSEED THETA PSEED
R( SSEED COS XSEED
R* PER-1 PSIZE SSEED
RA PERT PER-1 1.0
A* XD(I) R1 PERT
A* YD(I) R2 PERT
ND
VARIABLES
* Parameters of the ellipse
H11
H12
H22
G1
G2
DO I 1 NPTS
* Projections of the data points onto the ellipse
X X(I)
X Y(I)
ND
GROUPS
DO I 1 NPTS
XN OX(I) X(I) 1.0
XN OY(I) Y(I) 1.0
XE E(I)
ND
CONSTANTS
DO I 1 NPTS
Z ORTHREGC OX(I) XD(I)
Z ORTHREGC OY(I) YD(I)
X ORTHREGC E(I) 1.0
ND
BOUNDS
FR ORTHREGC 'DEFAULT'
START POINT
ORTHREGC H11 1.0
ORTHREGC H12 0.0
ORTHREGC H22 1.0
ORTHREGC G1 1.0
ORTHREGC G2 1.0
DO I 1 NPTS
Z ORTHREGC X(I) XD(I)
Z ORTHREGC Y(I) YD(I)
ND
ELEMENT TYPE
EV HXX H X
EV HXY H X
EV HXY Y
EV GX G X
ELEMENT USES
DO I 1 NPTS
XT EA(I) HXX
ZV EA(I) H H11
ZV EA(I) X X(I)
XT EB(I) HXY
ZV EB(I) H H12
ZV EB(I) X X(I)
ZV EB(I) Y Y(I)
XT EC(I) HXX
ZV EC(I) H H22
ZV EC(I) X Y(I)
XT ED(I) GX
ZV ED(I) G G1
ZV ED(I) X X(I)
XT EE(I) GX
ZV EE(I) G G2
ZV EE(I) X Y(I)
ND
GROUP TYPE
GV L2 GVAR
GROUP USES
DO I 1 NPTS
XT OX(I) L2
XT OY(I) L2
XE E(I) EA(I) EB(I) 2.0
XE E(I) EC(I) ED(I) -2.0
XE E(I) EE(I) -2.0
ND
OBJECT BOUND
LO ORTHREGC 0.0
* Solution
*LO SOLTN(10) 0.399058640
*LO SOLTN(50) 1.975569984
*LO SOLTN(250) 9.581964015
*LO SOLTN(500) 18.79064716
*LO SOLTN(2500) ???
*LO SOLTN(5000) ???
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS ORTHREGC
INDIVIDUALS
T HXX
F H * X * X
G H X * X
G X 2.0 * H * X
H H X X + X
H X X H + H
T HXY
F H * X * Y
G H X * Y
G X H * Y
G Y H * X
H H X Y
H H Y X
H X Y H
T GX
F G * X
G G X
G X G
H G X 1.0
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS ORTHREGC
* Least-square groups
INDIVIDUALS
T L2
F GVAR * GVAR
G GVAR + GVAR
H 2.0
ENDATA