EXAM BONUS

ES 204

Submitted by:

Timothy John S. Acosta

Submitted on:

April 22, 2015

Executive Summary: Adaptive Integration

A. Define the Problem

Integrate the function

100

2sin(

10

3

1

)

adaptively with a tolerance of 1x10-5

B. Problems Encountered

Working with multiple functions.

C. References

Gerald and Wheatley.(2004).Applied Numerical Analysis. 7th Ed. USA: Pearson

Education, Inc..

D. Results

Figure 1. Function plot with the intervals displayed

-80

-60

-40

-20

0

20

40

60

80

0 0.5 1 1.5 2 2.5 3 3.5

Integration from [1,3]

Interval Error Tolerance I

a b

1 1.00781 1.47E-08 3.91E-08 -0.3961

1.00781 1.01562 1.42E-08 3.91E-08 -0.33845

1.01562 1.02344 1.36E-08 3.91E-08 -0.28137

1.02344 1.03125 1.29E-08 3.91E-08 -0.2252

1.03125 1.03906 1.22E-08 3.91E-08 -0.17025

1.03906 1.04688 1.14E-08 3.91E-08 -0.11679

1.04688 1.05469 1.06E-08 3.91E-08 -0.06505

1.05469 1.0625 9.86E-09 3.91E-08 -0.01523

1.0625 1.07031 9.08E-09 3.91E-08 0.032519

1.07031 1.07812 8.32E-09 3.91E-08 0.078052

1.07812 1.08594 7.57E-09 3.91E-08 0.121267

1.08594 1.09375 6.86E-09 3.91E-08 0.162087

1.09375 1.10156 6.17E-09 3.91E-08 0.200454

1.10156 1.10938 5.52E-09 3.91E-08 0.236333

1.10938 1.11719 4.90E-09 3.91E-08 0.269709

1.11719 1.125 4.32E-09 3.91E-08 0.300583

1.125 1.13281 3.78E-09 3.91E-08 0.32897

1.13281 1.14062 3.27E-09 3.91E-08 0.3549

1.14062 1.14844 2.80E-09 3.91E-08 0.378414

1.14844 1.15625 2.37E-09 3.91E-08 0.399563

1.15625 1.17188 5.73E-08 7.81E-08 0.853416

1.17188 1.1875 3.61E-08 7.81E-08 0.91125

1.1875 1.20312 1.88E-08 7.81E-08 0.952718

1.20312 1.21875 4.95E-09 7.81E-08 0.979174

1.21875 1.23438 5.94E-09 7.81E-08 0.99203

1.23438 1.25 1.43E-08 7.81E-08 0.992716

1.25 1.26562 2.04E-08 7.81E-08 0.982647

1.26562 1.28125 2.48E-08 7.81E-08 0.963192

1.28125 1.29688 2.76E-08 7.81E-08 0.935661

1.29688 1.3125 2.93E-08 7.81E-08 0.901288

1.3125 1.32812 3.00E-08 7.81E-08 0.861223

1.32812 1.34375 2.99E-08 7.81E-08 0.816527

1.34375 1.35938 2.92E-08 7.81E-08 0.768167

1.35938 1.375 2.82E-08 7.81E-08 0.717022

1.375 1.39062 2.68E-08 7.81E-08 0.66388

1.39062 1.40625 2.52E-08 7.81E-08 0.609442

1.40625 1.42188 2.35E-08 7.81E-08 0.554327

1.42188 1.4375 2.17E-08 7.81E-08 0.499078

1.4375 1.45312 1.99E-08 7.81E-08 0.444164

1.45312 1.46875 1.81E-08 7.81E-08 0.38999

1.46875 1.48438 1.64E-08 7.81E-08 0.336897

1.48438 1.5 1.47E-08 7.81E-08 0.285172

1.5 1.51562 1.31E-08 7.81E-08 0.235052

1.51562 1.53125 1.17E-08 7.81E-08 0.186729

1.53125 1.54688 1.03E-08 7.81E-08 0.140353

1.54688 1.5625 9.02E-09 7.81E-08 0.096039

1.5625 1.57812 7.86E-09 7.81E-08 0.053871

1.57812 1.59375 6.80E-09 7.81E-08 0.013904

1.59375 1.60938 5.84E-09 7.81E-08 -0.02383

1.60938 1.625 4.97E-09 7.81E-08 -0.05932

1.625 1.65625 1.22E-07 1.56E-07 -0.21619

1.65625 1.6875 8.25E-08 1.56E-07 -0.33169

1.6875 1.71875 5.12E-08 1.56E-07 -0.43039

1.71875 1.75 2.72E-08 1.56E-07 -0.51326

1.75 1.8125 6.73E-08 3.13E-07 -1.21766

1.8125 1.84375 1.36E-08 1.56E-07 -0.67879

1.84375 1.875 2.00E-08 1.56E-07 -0.71046

1.875 1.90625 2.41E-08 1.56E-07 -0.73243

1.90625 1.9375 2.64E-08 1.56E-07 -0.74585

1.9375 1.96875 2.74E-08 1.56E-07 -0.75182

1.96875 2 2.74E-08 1.56E-07 -0.75132

2 2.03125 2.68E-08 1.56E-07 -0.7453

2.03125 2.0625 2.56E-08 1.56E-07 -0.73457

2.0625 2.09375 2.41E-08 1.56E-07 -0.71989

2.09375 2.125 2.25E-08 1.56E-07 -0.70194

2.125 2.15625 2.08E-08 1.56E-07 -0.68132

2.15625 2.1875 1.90E-08 1.56E-07 -0.65856

2.1875 2.21875 1.73E-08 1.56E-07 -0.63413

2.21875 2.25 1.56E-08 1.56E-07 -0.60843

2.25 2.28125 1.40E-08 1.56E-07 -0.58184

2.28125 2.3125 1.25E-08 1.56E-07 -0.55464

2.3125 2.34375 1.11E-08 1.56E-07 -0.52712

2.34375 2.375 9.87E-09 1.56E-07 -0.4995

2.375 2.4375 2.62E-07 3.13E-07 -0.91665

2.4375 2.5 2.01E-07 3.13E-07 -0.8092

2.5 2.5625 1.52E-07 3.13E-07 -0.70609

2.5625 2.625 1.13E-07 3.13E-07 -0.60852

2.625 2.6875 8.30E-08 3.13E-07 -0.51721

2.6875 2.75 5.94E-08 3.13E-07 -0.43252

2.75 2.8125 4.12E-08 3.13E-07 -0.35457

2.8125 2.875 2.75E-08 3.13E-07 -0.28327

2.875 3 4.18E-07 6.25E-07 -0.37814

Table 1. Intervals with corresponding integration

The integration over the domain using adaptive integration gave a value of -1.42603.

Appendix 1. List of Programs

double adaptintegration(double a,double b,double c,double x,double xT,double tol,double beg,double endd){ double simpsteps(double a,double b, double &c,double &x,double &xT,double &xa); double error,h=(b-a)/2; double xa=0; ofstream filer; filer.open("data.txt",ios_base::app); cout

double simps(double z,double y); double rxtrap(double k,double MA,double LA); double evaluate(double v); double x1,x2; //Large Interval xa=simps(a,b); //Bisection Interval c=a+(b-a)/2; x1=simps(a,c); x2=simps(c,b); xT=x1+x2; x=rxtrap(1,xT,xa); } double simps(double a,double b){ double evaluate(double z); double ans=0; double h=(b-a)/2; double c=a+h; a=evaluate(a); b=evaluate(b); c=evaluate(c); ans=h/3*(a+4*c+b); return ans; } double rxtrap(double k,double MA,double LA){ double ans; ans=(pow(4,k)*MA-LA)/(pow(4,k)-1); return ans; } double evaluate(double a){ double x; x=100/(a*a)*sin(10/a); return x; } int main() { double beg=1,endd=3; double x=0,xT=0; double error,a,b,c; double tol=1e-5;

//Initial Conds a=beg; b=endd; ans=adaptintegration(a,b,c,x,xT,tol,beg,endd); cout