Numerical integration

Transcription

1 Class Progress Basics of Linux, gnuplot, C Visualization of numerical data Roots of nonlinear equations (Midterm 1) Solutions of systems of linear equations Solutions of systems of nonlinear equations Monte Carlo simulation Interpolation of sparse data points Numerical integration (Midterm ) Solutions of ordinary differential equations 1

2 Integration of a Function from Known Data Points f(x) Integrand function x x x1 x x Known data points (xi,yi) from computation or measurement

3 Integration of a Function with Trapezoidal Rule f(x) x Approximate integral as sum of trapezoids x1 x x x

4 Sum the Area of Each Trapezoid f(x) y Area = (x - x1)(y1 + y)/ Sum up the trapezoidal areas to find final approximation to the integral y1 x coordinates do not have to be uniformly spaced x x1 x x x 4

5 Integration C Code for Tabulated Data Points int trapezoidal_data(int n,struct pointd *sample) { double integral; int i; integral =.; i = 1; while (i < n) { integral +=.5 * (sample[i].y + sample[i-1].y) * (sample[i].x sample[i-1].x); i++; } printf("%.1g\n",integral); return(); } 5

6 Trapezoidal Rule with Uniformly Spaced x Values f(x) Integrand function x x+h x+h x+h xn-1=x+4h x Define h = xi+1 - xi as the step size Four steps shown in this example 6

7 Integration of a Function with Trapezoidal Rule h Each trapezoidal area now = ( y i + y i+ 1 ) So sum up the areas of all the trapezoids: h h y + y 1 h h y 1+ y h h y + y... + h h y n + y n 1 h h y + h y 1 + h y + + h y n + y n 1 7

8 Integration of a Function with Trapezoidal Rule For n uniformly spaced data points: xn 1 x [ ( y + y n 1 ) n f ( x)dx h + yi i=1 ] 1 ( x n 1 x )h f ' ' (ξ) 1 ( ) Error term = 8

9 Simpson's Rule with Uniformly Spaced x Values f(x) Consider data point steps in consecutive pairs Fit a quadratic polynomial to three data points Fit a quadratic polynomial to three data points a a+h a+h Integrate under quadratic polynomial to find this area a+h b=a+4h x Integrate under quadratic polynomial to find this area 9

10 Integration of a Function with Simpson's 1/ Rule Fit second order Lagrange interpolation polynomial to first three data points: ( x x 1 )( x x ) ( x x )( x x ) ( x x )( x x 1 ) f ( x)= y + y 1+ y ( x x 1 )( x x ) ( x 1 x )( x1 x ) ( x x )( x x 1 ) Change x axis so that x= x1=h x=h and rewrite Lagrange interpolation polynomial for uniformly spaced data points: ( x h)( x h) x ( x h) x ( x h) f ( x)= y y 1+ y h h h Integrate under quadratic polynomial from to h: h h f ( x) dx ( ) [ y + 4 y1 + y ] 1

11 Integration of a Function with Simpson's 1/ Rule So sum up the areas under all the quadratic polynomials: h 4h h y + y 1+ y h 4h h y+ y + y 4 h 4h h y 4+ y 5+ y h 4h h y n + y n + y n 1 h 4h h 4h h h 4h h y+ y1+ y+ y+ y y n + y n + y n 1 11

12 Integration of a Function with Simpson's 1/ Rule So for n uniformly spaced data points: x n 1 x [ n n h f ( x) dx ( ) ( y + y n 1 ) + 4 y i + y i i=1, i odd i=,i even ] Applicable for only an even number of steps, so n-1 must be an even number, or the number of function points must be odd 1 ( x n 1 x )h 4 f (4) (ξ) 18 ( ) Error term = 1

13 Integration C Code for C Function #include <stdio.h> #include <stdlib.h> #include "integrate.h" int trapezoidal(double (*func)(double x),double xstart,double xstop,double xinc) { double x,integral; x = xstart; Note '-' sign to stop loop before the last desired point! integral =.5 * xinc * (*func)(x); x = x + xinc; xstop = xstop - (xinc *.5); while (((xinc >.) && (x < xstop)) ((xinc <.) && (x > xstop))) { integral += xinc * (*func)(x); x = x + xinc; } integral +=.5 * xinc * (*func)(x); printf("%.8g\n",integral); return(); } 1

14 Integration C Code for C Function int simpson(double (*func)(double x),double xstart,double xstop,double xinc) { double x,integral; int state; x = xstart; integral = xinc * (*func)(x) /.; x = x + xinc; xstop = xstop - (xinc *.5); state = ; while (((xinc >.) && (x < xstop)) ((xinc <.) && (x > xstop))) { if (state == ) { integral += 4. * xinc * (*func)(x) /.; state = 1; } else { integral +=. * xinc * (*func)(x) /.; state = ; } x = x + xinc; } if (state!= 1) { fprintf(stderr,"warning: Odd number of integration segments\n"); } integral += xinc * (*func)(x) /.; printf("%.8g\n",integral); return(); } 14

15 Integration C Code #include #include #include #include #include <stdio.h> <stdlib.h> <math.h> "integrate.h" "constants.h" double mu,sigma; double gaussian(double x) { return(exp(-(x - mu) * (x - mu) / (. * sigma * sigma)) / (sqrt(. * PI) * sigma)); } int main(int argc,char *argv[]) { double xstart,xstop,xinc; if (argc!= 6) { fprintf(stderr,"%s <mu> <sigma> <xstart> <xstop> <xinc>\n",argv[]); exit(1); } mu = atof(argv[1]); sigma = atof(argv[]); xstart = atof(argv[]); xstop = atof(argv[4]); xinc = atof(argv[5]); printf("trapezoidal rule:\n"); trapezoidal(gaussian,xstart,xstop,xinc); printf("simpson's rule:\n"); simpson(gaussian,xstart,xstop,xinc); exit(); } 15

16 Rotational Inertia For a given object, define I = r dm As the rotational inertia of an object about a particular axis of rotation Then simple rotational equivalent of the Newtonian translational relationships can be derived, such as Torque τ= I α 1 Kinetic energy K = I ω 16

17 Sample Rotational Inertia Integrals 17

18 Simple Rotational Inertial Problem dm = μrdθ r θ π Circular hoop Total mass = M Linear mass density = μ I = r dm= r (μ r d θ) Let r= R, a constant π I =R μ d θ= π R μ =( π R μ) R = M R 18

19 Not So Simple Rotational Inertial Problem dm = μrdθ r(θ) θ Not so circular hoop Total mass = M Linear mass density = μ π I = r (θ) dm= r (θ) (μ r (θ)d θ) π =μ r (θ)d θ 19

20 Rocket Propulsion Mass m Velocity v Initial state:

21 Rocket Propulsion Mass m+dm Final state, after small mass of fuel consumed and propelled out the nozzle is -dm: Velocity v+dv Mass -dm Velocity u 1

22 Rocket Propulsion Conservation of linear momentum: Pi= P f mv = dm u + (m+ dm)(v+ dv) Look at velocity of rocket relative to the exhaust in final state v rel v+ dv = v rel + u or u = v+ dv v rel Substituting for u : dm v rel = m dv Divide by dt to see thrust and acceleration of rocket: dm dv v rel = T = m dt dt

23 Rocket Propulsion Or integrate dv to find increase in rocket velocity over time: dm dv = v rel m vf mf i i dm dv = v rel m v m If v rel is constant mi v f v i = v rel ln ( ) mf But in general mf v rel (m) v f v i = dm m m i

24 Circumference of Elliptical Orbits a = Semi-major axis b = Semi-minor axis Comet in highly eccentric elliptical orbit b a Sun at one focus of ellipse b Eccentricity ϵ= 1 a 4

25 Circumference of Elliptical Orbits The circumference of an elliptical orbit C = 4 a E (ϵ) where a is the length of the semi-major axis, ϵ is the eccentricity and where the function E is the complete elliptic integral of the second kind: π E (ϵ) = 1 ϵ sin (θ)d θ This integral cannot be evaluated analytically. Note: for a circular orbit, a=b=radius, ϵ= and π E (ϵ) = π d θ = so C = 4 a π = π a as expected 5

26 Plotting Vectors with gnuplot We have used: plot 'problem1.dat' using 1: with lines plot 'problem.dat' using 1: with points Column with X coordinate Column with Y coordinate Another plot mode: plot 'helmholtz.dat' using 1:::4 with vectors Column with starting X coordinate Column with starting Y coordinate Column with X distance Column with Y distance 6

27 Electric Field Vectors for Dipole

28 Electric Field Vectors for Quadrapole

29 Electric Field Vectors for Example Arbitrary Charges

30 C Code for Mapping Electric Field Vectors xstop = xstop + xinc *.5; ystop = ystop + yinc *.5; rtest = 1.e-6 * (xinc*xinc + yinc*yinc); x = xstart; while (((xinc >.) && (x < xstop)) ((xinc <.) && (x > xstop))) { y = ystart; while (((yinc >.) && (y < ystop)) ((yinc <.) && (y > ystop))) { ex =.; ey =.; i = ; while (i < n) { rsq = (x-chg[i].x)*(x-chg[i].x) + (y-chg[i].y)*(y-chg[i].y); if (rsq < rtest) break; r = sqrt(rsq); Loop over n charges. ex += chg[i].q * (x-chg[i].x) / (r * rsq); Structure array chg[] holds ey += chg[i].q * (y-chg[i].y) / (r * rsq); each charge qi and i++; } xi and yi coordinates if (i >= n) { emag = sqrt(ex*ex + ey*ey); vx = xinc * ex / emag; vy = yinc * ey / emag; printf("%.8g %.8g %.8g %.8g\n",x-.5*vx,y-.5*vy,vx,vy); } y = y + yinc; } x = x + xinc; }

31 Classical Electric Field Lines Note that the vectors plotted by this method only record the direction of the E field vector at an array of sampled points. These are not quite equivalent to the classical field lines, which convey both the direction of the field and its magnitude from the density of drawn lines. 1

32 Helmholtz Coil for Uniform Magnetic Field

33 Magnetic Field of a Helmholtz Coil Coordinates of two conductors follow the parameterized equations x= R cos(θ) y=r sin(θ) R z=±d=± for π θ π R = coil radius D = z distance from D = R/ for best uniform fields z R current I D (Region of approximately uniform magnetic field in z direction) y D x current I

34 Finding Magnetic Field with Biot-Savart Law d B =d B 1 +d B probe at ( x, y, z ) z d l 1 r 1 r current I μ I d l i ri d Bi= 4π ri y B ( r )= d B d l x current I 4

35 Review of Vector Cross Product = A if C B then C x = A y Bz A z B y C y = A z B x A x Bz C z = A x B y A y Bx When computing d l r for this geometry, look for several components and terms that are zero! 5

36 Magnetic Field Along z-axis R=1m I=1A x= y= 1.e-6 Total Top coil Bottom coil z component of B field (T) 8.e-7 6.e-7 4.e-7.e-7.e z-coordinate (m) 6

37 Magnetic Field Off Of z-axis R=1m I=1A y= 1.e-6 x= x=.5m x=.1m x=.15m x=.m z component of B field (T) 8.e-7 6.e-7 4.e-7.e-7.e z-coordinate (m) 7

38 Vector Plot of Magnetic Field in Y-Z Plane Bottom Coil Only Z-coordinate Y-coordinate 8

39 Vector Plot of Magnetic Field in Y-Z Plane Top Coil Only Z-coordinate Y-coordinate 9

40 Vector Plot of Magnetic Field in Y-Z Plane Both Coils Z-coordinate Y-coordinate 4

41 Magnetic Field of a Real Solenoid B=μ i n where n=turns per unit length 41

42 Magnetic Field of a Real Solenoid N = number of turns H = height z R Coordinates of solenoid conductor follow the parameterized equations x=r cos(θ) y= R sin (θ) z=a θ H/ y for N π θ N π H/ and where a= H π N x current I 4

43 Finding Magnetic Field with Biot-Savart Law z d l = db r d B probe at ( x, y, z ) μ I d l r 4π r B ( r )= d B y x current I 4

44 Beware of Random Walks of Roundoff Error! Example: edit integration lab program to try to compute pi: 1 π= 4 1 x dx 44

45 Beware of Random Walks of Roundoff Error! #include #include #include #include <stdio.h> <stdlib.h> <math.h> "integrate.h" double func(double x) { return(4. * sqrt(1. - x*x)); } int main(int argc,char *argv[]) { double xinc; if (argc!= ) { fprintf(stderr,"%s <xinc>\n",argv[]); exit(1); } xinc = atof(argv[1]); printf("trapezoidal rule:\n"); trapezoidal(func,.,1.,xinc); printf("simpson's rule:\n"); simpson(func,.,1.,xinc); exit();} 45

46 Beware of Random Walks of Roundoff Error! $./pi_integrate.1 Trapezoidal rule: -nan Simpson's rule: -nan $./pi_integrate.5 Trapezoidal rule: Simpson's rule: Try two different step sizes through the integration routines So what is this nan business??? 46

47 NaN Binary Storage Convention 1. = s= e = 1-1 = q = 1. 1.=( 1) 1.. = 1. s=. by convention all actual bits NaN = s=1 all exponent bits =1 Most significant mantissa bit=1 NaN by convention 47

48 The man Page for sqrt() SQRT() NAME Linux Programmer's Manual SQRT() sqrt, sqrtf, sqrtl - square root function SYNOPSIS #include <math.h> double sqrt(double x); float sqrtf(float x); long double sqrtl(long double x); Link with -lm. DESCRIPTION The sqrt() function returns the nonnegative square root of x. RETURN VALUE!!! On success, these functions return the square root of x. If x is a NaN, a NaN is returned. If x is + (-), + (-) is returned. If x is positive infinity, positive infinity is returned. If x is less than -, a domain error occurs, and a NaN is returned. 48