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**CALL SPLINE(***fitted, data<, smooth><, delta><, nout>*

*<, type><, slope>***);**

**CALL SPLINEC(***fitted, coeff, endval, data<, smooth><, delta>*

*<, nout><, type><, slope>***);**

The SPLINEC subroutine returns the following values:

*fitted*- is an
*n*×2 matrix of fitted values. *coeff*- is an
*n*×5 (or*n*×9) matrix of spline coefficients. The matrix contains the cubic polynomial coefficients for the spline for each interval. Column 1 is the left endpoint of the*x*-interval for the regular (nonparametric) spline or the left endpoint of the parameter for the parametric spline. Columns 2-5 are the constant, linear, quadratic, and cubic coefficients, respectively, for the*x*-component. If a parametric spline is used, then columns 6-9 are the constant, linear, quadratic, and cubic coefficients, respectively, for the*y*-component. The coefficients for each interval are with respect to the variable*x*-*x*_{i}where*x*_{i}is the left endpoint of the interval and*x*is the point of interest. The matrix*coeff*can be processed to yield the integral or the derivative of the spline. This, in turn, can be used with the SPLINEV function to evaluate the resulting curves. The SPLINEC call returns*coeff*. *endval*- is a 1 ×2 matrix of endpoint values.
Slopes of the two ends of the curve are
reported as angles expressed in degrees.
The SPLINEC call returns the
*endval*argument.

The inputs to the SPLINEC subroutine are as follows:

*data*- specifies a
*n*×2 (or*n*×3) matrix of (*x*,*y*) points on which the spline is to be fit. The optional third column is used to specify a weight for each data point. If*smooth*>0, the weight column is used in calculations. A weight causes the data point to be ignored in calculations. *delta*- is an optional scalar specifying the resolution constant.
If
*delta*is specified, the fitted points are spaced by the amount*delta*on the scale of the first column of*data*if a regular spline is used or on the scale of the curve length if a parametric spline is used. If both*nout*and*delta*are specified,*nout*is used and*delta*is ignored. *nout*- is an optional scalar specifying the
number of fitted points to be computed.
The default is
*nout*=200. If*nout*is specified, then*nout*equally spaced points are returned. The*nout*argument overrides the*delta*argument. *slope*- is an optional 1 ×2 matrix of
endpoint slopes given as angles in degrees.
If a parametric spline is used, the
angle values are used modulo 360.
If a nonparametric spline is used, the tangent of
the angles is used to set the slopes (that is, the
effective angles range from -90 to 90 degrees).
*smooth*- is an optional scalar specifying
the degree of smoothing to be used.
If
*smooth*is omitted or set equal to 0, then a cubic interpolating spline is fit to the data. If*smooth*>0, then a cubic spline is used. Larger values of*smooth*generate more smoothing. *type*- is an optional 1 ×1 (or 1 ×2) character
matrix or quoted literal giving the type of spline to be used.
The first element of
*type*should be one of the following:-
`periodic`, which requests periodic endpoints -
`zero`, which sets second derivatives at endpoints to 0

The*type*argument controls the endpoint constraints unless the*slope*argument is specified. If`periodic`is specified, the response values at the beginning and end of column 2 of*data*must be the same unless the smoothing spline is being used. If the values are not the same, an error message is printed and no spline is fit. The default value is`zero`. The second element of*type*should be one of the following:-
`nonparametric`, which requests a nonparametric spline -
`parametric`, which requests a parametric spline

`parametric`is specified, a parameter sequence {*t*_{i}} is formed as follows:*t*=0 and_{1}*data*. The resulting splined values are paired to form the output. Changing the relative scaling of the first two columns of*data*changes the output because the sequence {*t*_{i}} assumes Euclidean distance.

Note that if the points are not arranged in ascending order by the first columns of*data*, then a parametric method must be used. An error message results if the nonparametric spline is requested. -

proc iml; data = { 0 1, 1 2, 2 3, 3 4, 4 5, 5 6, 6 7, 7 8, 8 9, 9 10 }; call splinec(fitted,coeff,endval,data,,,,'zero',{45 45}); v = splinev(coeff,{-1 1 2 3 3.5 4 20}); print v; v = splinev(coeff,,3); print v;

This example takes a function defined by discrete data and finds the integral and the first moment of the function.

data func; input x @@; y = x+0.1*sin(x); cards; 0 2 5 7 8 10 ; proc iml; use func; read all into a; call splinec(fit,coeff,endval,a,,,,'zero'); start fcheck(x) global(coeff,pow); /************************************/ /* Note that the first column of v */ /* contains the points of */ /* evaluation and the second column */ /* contains the evaluation. */ /************************************/ v = x##pow # splinev(coeff,x //x)[1,2]; return(v); finish; start moment(po) global(coeff,pow); pow = po; call quad(z,'fcheck',coeff[,1]) eps = 1.e-10; v1 = sum(z); return(v1); finish; mass = moment(0); /* to compute the mass */ mass = mass // (moment(1)/mass) // /* to compute the mean */ (moment(2)/mass) ; /* to compute the meansquare */ print mass; /*****************************************************/ /* Use Gauss-Legendre integration: this is not */ /* adaptive, but it is good for moments up to maxng. */ /*****************************************************/ gauss = { -9.3246951420315205e-01 -6.6120938646626448e-01 -2.3861918608319743e-01 2.3861918608319713e-01 6.6120938646626459e-01 9.3246951420315183e-01, 1.713244923791701e-01 3.607615730481388e-01 4.679139345726905e-01 4.679139345726904e-01 3.607615730481389e-01 1.713244923791707e-01 }; ngauss = ncol(gauss); maxng = 2*ngauss-4; start moment1(pow) global(coeff,gauss,ngauss,maxng); if pow < maxng then do; nrow = nrow(coeff); ncol = ncol(coeff); left = coeff[1:nrow-1,1]; right = coeff[2:nrow,1]; mid = 0.5*(left+right); interv = 0.5*(right - left); /* scale the weights on each interval */ wgts = shape(interv*gauss[2,],1); /* scale the points on each interval */ pts = shape(interv*gauss[1,] + mid * repeat(1,1,ngauss),1) ; /* evaluate the function */ eval = splinev(coeff,pts)[,2]`; mat = sum (wgts#pts##pow#eval); end; return(mat); finish; mass = moment1(0); /* to compute the mass */ mass = mass // (moment1(1)/mass) // (moment1(2)/mass) ; print mass;

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