.p>.CMCH3.DOC!USER_FCN_LEAST_SQUARES;user_fcn_least_squares
Computes a least-squares fit using user-supplied functions.
Synopsis
#include <imsl.h>
float *imsl_f_user_fcn_least_squares (float fcn (int k, float x), int nbasis, int ndata, float xdata[], float ydata[], ¼, 0)
The type double function is imsl_d_user_fcn_least_squares.
Required Arguments
float fcn (int k, float x)
(Input)
User-supplied function that defines the subspace from which the
least-squares fit is to be performed. The k-th basis function evaluated
at x is f(k, x) where k
= 1, 2,
¼, nbasis.
int nbasis
(Input)
Number of basis functions.
int ndata
(Input)
Number of data points.
float xdata[]
(Input)
Array with ndata components
containing the abscissas of the least-squares problem.
float ydata[]
(Input)
Array with ndata components
containing the ordinates of the least-squares problem.
Return Value
A pointer to the vector containing the coefficients of the basis functions. If a fit cannot be computed, then NULL is returned. To release this space, use free.
Synopsis with Optional Arguments
#include <imsl.h>
float
*imsl_f_user_fcn_least_squares (), int
nbasis,
int
ndata,
float
xdata[],
float
ydata[],
IMSL_RETURN_USER, float
coef[],
IMSL_INTERCEPT, float
*intercept,
IMSL_SSE, float
*ssq_err,
IMSL_WEIGHTS, float
weights[],
IMSL_FCN_W_DATA, float fcn
( ), void *data,
0)
Optional Arguments
IMSL_RETURN_USER, float coef[]
(Output)
The coefficients are stored in the user-supplied array.
IMSL_INTERCEPT, float
*intercept (Output)
This option adds an intercept to the
model. Thus, the least-squares fit is computed using the user-supplied basis
functions augmented by the constant function. The coefficient of the constant
function is stored in intercept.
IMSL_SSE, float *ssq_err
(Output)
This option returns the error sum of squares.
IMSL_WEIGHTS, float weights[]
(Input)
This option requires the user to provide the weights.
Default:
all weights equal one
IMSL_FCN_W_DATA, float fcn (int k, float x, float *data),
void
*data, (Input)
User supplied function that defines the
subspace from which the least-squares fit is to be performed, which also accepts
a pointer to data that is supplied by the user. data is a pointer to
the data to be passed to the user-supplied function. See the Introduction, Passing Data to
User-Supplied Functions at the beginning of this manual for more
details.
Description
The function imsl_f_user_fcn_least_squares computes a best least-squares approximation to given univariate data of the form
by M basis functions
(where M = nbasis). In particular, the default for this function returns the coefficients a which minimize
where w = weights, n = ndata, x = xdata, and f = ydata.
If the optional argument IMSL_INTCERCEPT is chosen, then an intercept is placed in the model, and the coefficients a, returned by imsl_f_user_fcn_least_squares, minimize the error sum of squares as indicated below.
Examples
Example 1
This example fits the following two functions (indexed by δ):
1 + sinx + 7 sin3x + δe
where ɛ is a random uniform deviate over the range [−1, 1] and δ is 0 for the first function and 1 for the second. These functions are evaluated at 90 equally spaced points on the interval [0, 6]. Four basis functions are used: 1, sinx, sin2x, sin3x.
#include <imsl.h>
#include
<stdio.h>
#include <math.h>
#define NDATA
90
/* Define function */
#define F(x) (float)(1.+
sin(x)+7.*sin(3.0*x))
float
fcn(int n, float x);
main()
{
int nbasis = 4, i,
delta;
float
ydata[NDATA], xdata[NDATA], *random,
*coef;
/* Generate random numbers */
imsl_random_seed_set(1234567);
random =
imsl_f_random_uniform(NDATA, 0);
/*
Set up data */
for(delta = 0; delta < 2;
delta++) {
for (i = 0; i <
NDATA; i++)
{
xdata[i]
= 6.*(float)i
/((float)(NDATA-1));
ydata[i] = F(xdata[i]) +
(delta)*2.*(random[i]-.5);
}
coef =
imsl_f_user_fcn_least_squares(fcn, nbasis, NDATA, xdata,
ydata, 0);
printf("\nFor delta =
%1d", delta);
imsl_f_write_matrix("the computed coefficients
are\n",
1, nbasis,
coef, 0);
}
}
float fcn(int n, float
x)
{
return (n == 1) ? 1.0 :
sin((n-1)*x);
}
Output
For delta = 0
the computed coefficients
are
1
2
3
4
1
1
-0 7
For
delta = 1
the computed
coefficients are
1
2
3
4
0.979
0.998
0.096 6.839
Example 2
Recall that the first example fitted the following two functions (indexed by δ):
1 + sinx + 7 sin3x + δe
where ɛ is a random uniform deviate over the range[−1, 1] , and δ is 0 for the first function and 1 for the second. These functions are evaluated at 90 equally spaced points on the interval [0, 6]. Previously, the four basis functions were used: 1, sinx, sin2x, sin3x. This example uses the four basis functions: sinx, sin2x, sin3x, sin4x, combined with the intercept option.
#include <imsl.h>
#include
<stdio.h>
#include <math.h>
#define NDATA
90
/* Define function */
#define F(x) (float)(1.+
sin(x)+7.*sin(3.0*x))
float
fcn(int n, float x);
main()
{
int nbasis = 4, i,
delta;
float
ydata[NDATA], xdata[NDATA], *random, *coef,
intercept;
/* Generate random numbers
*/
imsl_random_seed_set(1234567);
random = imsl_f_random_uniform(NDATA,
0);
/* Set up data */
for(delta = 0; delta < 2;
delta++){
for (i = 0; i <
NDATA; i++) {
xdata[i] = 6.*(float)i
/((float)(NDATA-1));
ydata[i] = F(xdata[i]) +
(delta)*2.*(random[i]-.5);
}
coef =
imsl_f_user_fcn_least_squares(fcn, nbasis, NDATA, xdata,
ydata,
IMSL_INTERCEPT,
&intercept,
0);
printf("\nFor delta = %1d\n",
delta);
printf("The predicted
intercept value is %10.3f\n" ,
intercept);
imsl_f_write_matrix("the computed coefficients
are\n",
1, nbasis, coef, 0);
}
}
float fcn(int
n, float x)
{
return sin(n*x);
}
Output
For delta = 0
The predicted intercept value
is
1.000
the computed
coefficients are
1
2
3
4
1
0
7 -0
For delta =
1
The predicted intercept value is
0.978
the computed
coefficients are
1
2
3
4
0.998
0.097
6.841 0.075
Warning Errors
IMSL_LINEAR_DEPENDENCE Linear dependence of the basis functions exists. One or more components of coef are set to zero.
IMSL_LINEAR_DEPENDENCE_CONST Linear dependence of the constant function and basis functions exists. One or more components of coef are set to zero.
Fatal Errors
IMSL_NEGATIVE_WEIGHTS_2 All weights must be greater than or equal to zero.
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