Quadrature methods in Java assignment help

Newton Cotes, Gaussian, and quadrature methods in Java assignment help

The assignment deals with evaluating the closed Newton Cotes, Gaussian, and quadrature methods using 3 different functions, it displays the quadrature results, weights, and nodes used on the console. Below is Newton Cotes, Gaussian, and quadrature methods demonstrated by our Java assignment help providers.

Quadrature Results, Weights, and Nodes

Newton Cotes, Gaussian, and quadrature methods in Java assignment help


/*
* PURPOSE: Evaluatestheclosed Newton Cotes, Gaussian and Lobatto
* quadraturemethodsusing 3 differentfunctions, displaysthequadrature
* results, weights and nodesused
* OUTPUT: Printsallquadratureresultsonthescreen.
*/
publicclassQuadrature {
/**
* PURPOSE: Definitionofthefirstfunctionto use in thequadratures
* INPUT: x, position in X usedtoevaluatefunction
* OUTPUT: double, evaluationofthefirstfunction in X.
*/
publicstaticdouble function1(double x) {
return 1 - Math.sin(1 - x);
}
/**
* PURPOSE: Definitionofthesecondfunctionto use in thequadratures
* INPUT: x, position in X usedtoevaluatefunction
* OUTPUT: double, evaluationofthefirstfunction in X.
*/
publicstaticdouble function2(double x) {
returnMath.sqrt(x + 1) + 1;
}
/**
* PURPOSE: Definitionofthethirdfunctionto use in thequadratures
* INPUT: x, position in X usedtoevaluatefunction
* OUTPUT: double, evaluationofthefirstfunction in X.
*/
publicstaticdouble function3(double x) {
returnMath.tanh(x + 1);
}
/**
* PURPOSE: Evaluatethequadratureofthe 3 functionsdefinedaboveusing
* theclosed Newton Cotes quadrature
* INPUT: None
* OUTPUT: printsthequadratureresultsonthescreen.
*/
publicstaticvoidnewtonCotes() {
int n = 5; // use 5 points
double x0 = -1; // lowerlimit
doublexn = 1; // upperlimit
double h = (xn - x0)/n; // step size
double w [] = { // weights
19./288.,
25./96.,
25./144.,
25./144.,
25./96.,
19./288.
};
double x [] = new double[n+1]; // node positions
doublequadrature [] = {0.0, 0.0 ,0.0}; // itwillholdthequadratureresults
doubletrueArea[] = {
0.58385316345285761300, // true areaforfunction 1
3.8856180831641267317, // true areaforfunction 2
1.3250027473578644309 // true areaforfunction 3
};
// node positions
for (int i=0; i<=n; i++)
x[i] = x0 + i*h;
// Printweights and nodes
System.out.println("------------------------------------------------------------------");
System.out.println("5-point Closed Newton Cotes quadrature\n");
System.out.println("Weights and nodes:");
for (int i=0; i<=n; i++) {
System.out.printf("w[%d] = %f, x[%d] = %f\n", i, w[i], i, x[i]);
}
// Evaluatethequadratureoffirstfunctionusingthe 5-point rule
quadrature[0] = 0.0; // initializetozeroformakingthe sum
for (int i=0; i<=n; i++)
quadrature[0] += (xn - x0)*w[i]*function1(x[i]); //addnodecontributionto sum
// Evaluatethequadratureofsecondfunction
quadrature[1] = 0.0; // initializetozeroformakingthe sum
for (int i=0; i<=n; i++)
quadrature[1] += (xn - x0)*w[i]*function2(x[i]); //addnodecontributionto sum
// Evaluatethequadratureofthirdfunction
quadrature[2] = 0.0; // initializetozeroformakingthe sum
for (int i=0; i<=n; i++)
quadrature[2] += (xn - x0)*w[i]*function3(x[i]); //addnodecontributionto sum
System.out.println("Function\tAreaapproximation\tTruevalue\tError");
for (int i=0; i<3; i++) {
double error = quadrature[i] - trueArea[i];
System.out.printf("%d\t\t%f\t\t%f\t%f\n", i+1, quadrature[i], trueArea[i], error);
}
System.out.println();
}
/**
* PURPOSE: Evaluatethequadratureofthe 3 functionsdefinedaboveusing
* the Gaussian quadrature
* INPUT: None
* OUTPUT: printsthequadratureresultsonthescreen.
*/
publicstaticvoidgaussian() {
int n = 5; // use 5 points
double x0 = -1; // lowerlimit
doublexn = 1; // upperlimit
double w [] = { // weights
(322 - 13*Math.sqrt(70))/900.,
(322 + 13*Math.sqrt(70))/900.,
128./255.,
(322 + 13*Math.sqrt(70))/900.,
(322 - 13*Math.sqrt(70))/900.,
};
double x [] = { // node positions
-Math.sqrt(5 + 2*Math.sqrt(10./7.))/3.,
-Math.sqrt(5 - 2*Math.sqrt(10./7.))/3.,
0,
Math.sqrt(5 - 2*Math.sqrt(10./7.))/3.,
Math.sqrt(5 + 2*Math.sqrt(10./7.))/3.,
};
doublequadrature [] = {0.0, 0.0 ,0.0}; // itwillholdthequadratureresults
doubletrueArea[] = {
0.58385316345285761300, // true areaforfunction 1
3.8856180831641267317, // true areaforfunction 2
1.3250027473578644309 // true areaforfunction 3
};
// Printweights and nodes
System.out.println("------------------------------------------------------------------");
System.out.println("5-point Gaussian quadrature\n");
System.out.println("Weights and nodes:");
for (int i=0; i