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ORTHREGA.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME ORTHREGA
* Problem :
* *********
* An orthogonal regression problem.
* The problem is to fit (orthogonally) an ellipse to a set of points
* in the plane.
* 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.
* classification QQR2-AN-V-V
* Number of levels in the generation of the data points
* ( number of data points = 4**LEVELS
* number of variables = 2 * 4**LEVELS + 5
* number of constraints = 4**LEVELS )
*IE LEVELS 3 $-PARAMETER n = 133 original value
*IE LEVELS 4 $-PARAMETER n = 517
*IE LEVELS 5 $-PARAMETER n = 2053
IE LEVELS 6 $-PARAMETER n = 8197
* Parameters for the generation of the data points
RE A 9.0
RE B 6.0
RE CX 0.5
RE CY 0.5
* Constants
IE 1 1
RE PI 3.1415926535
* Computed parameters
RM -A A -1.0
RM -B B -1.0
* Construct the data points
IE NPTS 1
A= XD(1) CX
A= YD(1) CY
DO I 1 LEVELS
IA NP NPTS 0
DO J 1 NP
A= XZ(J) XD(J)
A= YZ(J) YD(J)
OD J
IE NPTS 0
DO J 1 NP
IA NPTS NPTS 1
A+ XD(NPTS) XZ(J) A
A+ YD(NPTS) YZ(J) A
IA NPTS NPTS 1
A+ XD(NPTS) XZ(J) B
A+ YD(NPTS) YZ(J) -B
IA NPTS NPTS 1
A+ XD(NPTS) XZ(J) -A
A+ YD(NPTS) YZ(J) -A
IA NPTS NPTS 1
A+ XD(NPTS) XZ(J) -B
A+ YD(NPTS) YZ(J) B
OD J
R/ A A PI
R/ B B PI
R/ -A -A PI
R/ -B -B PI
ND
VARIABLES
* Parameters of the ellipse
H11
H12
H22
G1
G2
* Projections of the data points onto the ellipse
DO I 1 NPTS
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 ORTHREGA OX(I) XD(I)
Z ORTHREGA OY(I) YD(I)
X ORTHREGA E(I) 1.0
ND
BOUNDS
FR ORTHREGA 'DEFAULT'
START POINT
ORTHREGA H11 1.0
ORTHREGA H12 0.0
ORTHREGA H22 1.0
ORTHREGA G1 0.0
ORTHREGA G2 0.0
DO I 1 NPTS
Z ORTHREGA X(I) XD(I)
Z ORTHREGA 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 ORTHREGA 0.0
* Solution
*LO SOLTN(3) 350.29936756
*LO SOLTN(4) 1414.0524915
*LO SOLTN(5) ???
*LO SOLTN(6) ???
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS ORTHREGA
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 ORTHREGA
* Least-square groups
INDIVIDUALS
T L2
F GVAR * GVAR
G GVAR + GVAR
H 2.0
ENDATA