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ORTHRGDM.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME ORTHRGDM
* Problem :
* *********
* An orthogonal regression problem.
* The problem is to fit (orthogonally) a circle to a set of points
* in the plane. This set of points is generated by perturbing a
* first set lying exactly on a predefined circle centered at the
* origin.Each point mapped to the center of the circle has a constraint
* gradient of zero, making the Jacobian of the constraints singular.
*
* The start point is modified here to keep any data points from
* converging to the center.
*
* 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, Mar 1991 and T. Plantenga, Sep 1993.
* classification QOR2-AY-V-V
* Number of data points
* (number of variables = 2 NPTS + 3 )
*IE NPTS 10 $-PARAMETER n = 23
*IE NPTS 50 $-PARAMETER n = 103
*IE NPTS 76 $-PARAMETER n = 155
*IE NPTS 100 $-PARAMETER n = 203
*IE NPTS 250 $-PARAMETER n = 503
*IE NPTS 500 $-PARAMETER n = 1003
*IE NPTS 1000 $-PARAMETER n = 2003
*IE NPTS 2000 $-PARAMETER n = 4003 original value
*IE NPTS 2500 $-PARAMETER n = 5003
IE NPTS 5000 $-PARAMETER n = 10003
* True cardioid circle (centered at the origin)
RE TZ3 1.7
* 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* Z3SQ TZ3 TZ3
RA 1+TZ3SQ Z3SQ 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
RM TDP_CT CT 0.25
R+ FACT 1+TZ3SQ TDP_CT
R* R1 FACT CT
R* R2 FACT ST
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 circle
Z1
Z2
Z3
DO I 1 NPTS
* Projections of the data points onto the circle
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
ZN ORTHRGDM OX(I) XD(I)
ZN ORTHRGDM OY(I) YD(I)
ND
BOUNDS
FR ORTHRGDM 'DEFAULT'
START POINT
ORTHRGDM Z1 1.0
ORTHRGDM Z2 0.0
ORTHRGDM Z3 1.0
DO I 1 NPTS
Z ORTHRGDM X(I) XD(I)
Z ORTHRGDM Y(I) YD(I)
ND
ELEMENT TYPE
EV TA X Y
EV TA ZA ZB
IV TA DX DY
EV TB X Y
EV TB ZA ZB
EV TB ZC
IV TB DX DY
IV TB ZZ
ELEMENT USES
DO I 1 NPTS
XT EA(I) TA
ZV EA(I) X X(I)
ZV EA(I) Y Y(I)
ZV EA(I) ZA Z1
ZV EA(I) ZB Z2
XT EB(I) TB
ZV EB(I) X X(I)
ZV EB(I) Y Y(I)
ZV EB(I) ZA Z1
ZV EB(I) ZB Z2
ZV EB(I) ZC Z3
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) -1.0
ND
OBJECT BOUND
LO ORTHRGDM 0.0
* Solution
*LO SOLTN(10) 3.412121065
*LO SOLTN(50) 15.59042181
*LO SOLTN(250) 76.10435792
*LO SOLTN(500) 151.2351183
*LO SOLTN(1000) 304.899
*LO SOLTN(2500) ???
*LO SOLTN(5000) ???
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS ORTHRGDM
TEMPORARIES
R T
R T1
R T1SQ
R ZZSQ
INDIVIDUALS
T TA
R DX X 1.0 ZA -1.0
R DY Y 1.0 ZB -1.0
A T DX * DX + DY * DY
F T * T
G DX 4.0 * T * DX
G DY 4.0 * T * DY
H DX DX 4.0 * ( T + 2.0 * DX * DX )
H DX DY 8.0 * DX * DY
H DY DY 4.0 * ( T + 2.0 * DY * DY )
T TB
R DX X 1.0 ZA -1.0
R DY Y 1.0 ZB -1.0
R ZZ ZC 1.0
A T DX * DX + DY * DY
A ZZSQ ZZ * ZZ
A T1 1.0 + ZZSQ
A T1SQ T1 * T1
F T * T1SQ
G DX 2.0 * DX * T1SQ
G DY 2.0 * DY * T1SQ
G ZZ 4.0 * T * T1 * ZZ
H DX DX 2.0 * T1SQ
H DX ZZ 8.0 * DX * T1 * ZZ
H DY DY 2.0 * T1SQ
H DY ZZ 8.0 * DY * T1 * ZZ
H ZZ ZZ 4.0 * T * ( 2.0 * ZZSQ + T1)
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS ORTHRGDM
* Least-square groups
INDIVIDUALS
T L2
F GVAR * GVAR
G GVAR + GVAR
H 2.0
ENDATA