Function Approximations in Java

FunctionApproximations.java

A class containing methods to calculate the coefficients and values based on a set of initial values.

Supports:

• Newton polynomial
• Splines of 2nd and 3rd order
• Least squares

Example: Approximating sine values based on 12 initial points.

int n = 11;

double[] stx = new double[n + 1];

double[] y = new double[n + 1];

double dif = (Math.PI * 2) / (n);

double start = -Math.PI;

for (int i = 0; i < n + 1; i++) {

    stx[i] = start;

    start += dif;

    y[i] = (double) Math.round(Math.sin(stx[i]) * 100000) / 100000;

}

double[] newtonPolCoef = FunctionApproximations.newtonPolCoef(stx, y);

double[] splineCoef = FunctionApproximations.splineCoef(stx, y, 3);

double[] lsCoef = FunctionApproximations.leastSqCoeff(stx, y, 5);

Random random=new Random();

double val,correct,newton,spline,leastsq;

for (int i = 0; i < 10; i++) {

    val=random.nextDouble();

    correct=Math.sin(val);

    newton=FunctionApproximations.newtonPolVal(newtonPolCoef, stx, getX(val));

    spline=FunctionApproximations.splineVal(splineCoef, stx, getX(val));

    leastsq=FunctionApproximations.leastSqVal(lsCoef, getX(val));

    System.out.println("sin("+val+")");

    System.out.println("Newton approximation :     "+newton+"    off by   "+(correct-newton));

    System.out.println("Cubic splines :            "+spline+"    off by   "+(correct-spline));

    System.out.println("Least squares 5th degree : " +leastsq+"    off by   "+(correct-leastsq));

    System.out.println("----");

}

static private double getX(double x) {

    if (Math.abs(x) <= Math.abs(Math.PI)) {

        return x;

    } else {

        if (x > 0) {

            x -= Math.round(x / (2 * Math.PI)) * (2 * Math.PI);


        } else {

            x += Math.round(Math.abs(x) / (2 * Math.PI)) * (2 * Math.PI);

        }

        return x;

    }

}

Sample output:

sin(0.8481863801681208)
Newton approximation :     0.7500825572784057    off by   -3.455657819895208E-7
Cubic splines :            0.7500853256850928    off by   -3.113972469104276E-6
Least squares 5th degree : 0.7446845168230829    off by   0.005397694889540783
----
sin(0.17370422573181987)
Newton approximation :     0.1728302183995918    off by   1.7900743105314643E-6
Cubic splines :            0.17285859735632642    off by   -2.6588882424072313E-5
Least squares 5th degree : 0.1704451410039368    off by   0.0023868674699655534
----
sin(0.9222664997998143)
Newton approximation :     0.7969736441928723    off by   -9.775787263022195E-7
Cubic splines :            0.7969127151051372    off by   5.995150900883761E-5
Least squares 5th degree : 0.7921460535443667    off by   0.004826613069779273
----
sin(0.42918641383160216)
Newton approximation :     0.4161283242433726    off by   2.818337875132304E-6
Cubic splines :            0.41601147561349894    off by   1.1966696774878827E-4
Least squares 5th degree : 0.4110357193164086    off by   0.005095423264839138
----
sin(0.5240316104273179)
Newton approximation :     0.5003722902694624    off by   2.5088399099315595E-6
Cubic splines :            0.5002095950629291    off by   1.6520404644326803E-4
Least squares 5th degree : 0.4946962081563806    off by   0.0056785909529917244
----
sin(0.9800507235185272)
Newton approximation :     0.8305269746540256    off by   -1.3510869794064462E-6
Cubic splines :            0.8303843741206196    off by   1.4124944642657233E-4
Least squares 5th degree : 0.8262637304226693    off by   0.00426189314437686
----
sin(0.9303877316203922)
Newton approximation :     0.8018527171875109    off by   -1.0374269545643244E-6
Cubic splines :            0.8017809837980641    off by   7.069596249231758E-5
Least squares 5th degree : 0.7970983401665966    off by   0.004753339593959738
----
sin(0.6236951478334793)
Newton approximation :     0.5840367621157957    off by   1.826036797991648E-6
Cubic splines :            0.5838957413617496    off by   1.428467908440867E-4
Least squares 5th degree : 0.5780582273554796    off by   0.005980360797114059
----
sin(0.5844538429625159)
Newton approximation :     0.5517418299721984    off by   2.1323006360596253E-6
Cubic splines :            0.5515834663608791    off by   1.6049591195543833E-4
Least squares 5th degree : 0.5458425471133572    off by   0.005901415159477286
----
sin(0.37727308863399756)
Newton approximation :     0.36838388119885634    off by   2.8261127495432525E-6
Cubic splines :            0.36831036229846276    off by   7.634501314313091E-5
Least squares 5th degree : 0.36372091601169265    off by   0.004665791299913236
----