1. Computational Physics Numerical Differentiation Dr. Guy Tel-Zur Clouds. Picture by Peter Griffin, publicdomainpictures.net

2. MHJ Chapter 3: Numerical Differentiation Should be f’c “2” stands for two points

3. forward/backward 1 st derivative: ±h f(x)=a+bx^2f(x)=a+bx f’=2bxf’=bExact f’ f 2 ’=((a+b(x+h)^2)-(a+bx^2))/h = (a+bx^2+2bxh+bh^2)-(a+bx^2))/h= 2bx+bh Bad!!! f 2 ’=((a+b(x+h))-(a+bx))/h=b Good! f2’f2’ ((a+b(x+h)^2)-(a+b(x-h)^2))/(2h)= (a+bx^2+2bxh+bh^2-a-bx^2+2bxh- bh^2)/(2h)=4bxh/2h=2bx Good! (1/2)(f 2L ’+f 2R ’)= f 2 ’=(f(x+h)-f(x-h))/(2h)

4. N-points stencil

5. Example code 2 nd derivative of exp(x) Code in C++, we will learn more of the language features: – Pointers – Call by Value/Reference – Dynamic memory allocation – Files (I/O)

6. Call by value vs. call by reference printf(“speed= %d\n”, v); // this is a call by value as the value of v won’t be changed by the function (printf) – which is desired scanf(“%d\n”,&v); // this is a call by reference, we want to supply the address of v in order to set it’s value(s)

7. // // This program module // demonstrates memory allocation and data transfer in between functions in C++ #include int main(int argc, char *argv[]) { int a; // line 1 int *b; // line 2 a = 10; // line 3 b = new int[10]; // line 4 for(i = 0; i < 10; i++) { b[i] = i; // line 5 } func( a,b); // line 6 return 0; } // End: function main() void func( int x, int *y) // line 7 { x += 7; // line 8 *y += 10; // line 9 y[6] += 10; // line 10 return; // line 11 } // End: function func() ניתוח תכנית לדוגמה

8. // // This program module // demonstrates memory allocation and data transfer in between functions in C++ #include void func( int x, int *y); int main(int argc, char *argv[]) { int a; // line 1 int *b; // line 2 a = 10; // line 3 b = new int[10]; // line 4 for(int i = 0; i < 10; i++) { b[i] = i; // line 5 } func( a,b); // line 6 printf("a=%d\n",a); printf("b[0]=%d\n",b[0]); printf("b[1]=%d\n",b[0]); printf("b[2]=%d\n",b[0]); printf("b[3]=%d\n",b[0]); printf("b[4]=%d\n",b[0]); printf("b[5]=%d\n",b[0]); printf("b[6]=%d\n",b[0]); printf("b[7]=%d\n",b[0]); printf("b[8]=%d\n",b[0]); printf("b[9]=%d\n",b[0]); return 0; } // End: function main() void func( int x, int *y) // line 7 { x += 7; // line 8 *y += 10; // line 9 y[6] += 10; // line 10 return; // line 11 } // End: function func() תכנית משופרת Check program: demo1.cpp Under /lectures/02/code Check program: demo1.cpp Under /lectures/02/code

9. הנושאים שיידונו מכאן הלאה עד לסיום פרק 3 מתוך MHJ תכנית מס ' 1: חישוב הנגזרת השנייה של exp(x) בשפת C++ תכנית מס ' 2: תכנית דומה בשפת C בתוספת התיחסות לפתיחת קבצים לקריאה ולכתיבה תכנית מס ' 3: כמו תכנית מס ' 2 אלא שהפעם בשפת C++ והדגשת השוני בינהן. תכנית מס ' 4: כנ " ל בשפת Fortran90 דיון בהערכת השגיאה בחישוב הצגה גראפית של השגיאה – תוכנת gnuplot

10. Self.Open( DevC++ for the demos!) תזכורת לעצמי I slightly modified “program1.cpp” from MHJ section 3.2.1 (2009 Fall edition): http://www.fys.uio.no/compphys/cp/program s/FYS3150/chapter03/cpp/program1.cpp http://www.fys.uio.no/compphys/cp/program s/FYS3150/chapter03/cpp/program1.cpp

11. Explain program1.cpp Working directory: C:\Users\telzur\Documents\BGU\Teaching\ComputationalPhysics\2011A\Lectures\02\code> Open DevC++ IDE for the demo Usage: > program1 0.01 10 100 Examine the output: > type out.dat

12. program2.cpp The book mentions program2.cpp which is in cpp and the URL is indeed a cpp code, but the listing below the URL is in C. This demonstrates the I/O differences between C and C++

13. using namespace std; #include int main(int argc, char *argv[]) { FILE *in_file, *out_file; if( argc < 3) { printf("The programs has the following structure :\n"); printf("write in the name of the input and output files \n"); exit(0); } in_file = fopen( argv[1], "r"); // returns pointer to the input file if( in_file == NULL ) { // NULL means that the file is missing printf("Can't find the input file %s\n", argv[1]); exit(0); } out_file = fopen( argv[2], "w"); // returns a pointer to the output file if( out_file == NULL ) { // can't find the file printf("Can't find the output file%s\n", argv[2]); exit(0); } fclose(in_file); fclose(out_file); return 0; } Working with files in C++ program2.cpp Working with files in C++ program2.cpp

14. program3.cpp Usage: > program3 outfile_name All the rest is like in program1.cpp

15. Now lets check the f90 version Open in SciTE program1.f90 In the image below: compilation and execution demo:

16. MHJ section3.2.2 Error analysis

17. Content of exp(10)’’ computation See MHJ section 3.2.2 and Fig. 3.2 (Fall 2009 Edition) Text output with 4 columns: h, computed_derivative, log(h),ε >program1 Input: 0.1 10. 10 >more out.dat 1.00000E-001 2.72055E+000 -1.00000E+000 -3.07904E+000 5.00000E-002 2.71885E+000 -1.30103E+000 -3.68121E+000 2.50000E-002 2.71842E+000 -1.60206E+000 -4.28329E+000 1.25000E-002 2.71832E+000 -1.90309E+000 -4.88536E+000 6.25000E-003 2.71829E+000 -2.20412E+000 -5.48742E+000 3.12500E-003 2.71828E+000 -2.50515E+000 -6.08948E+000 1.56250E-003 2.71828E+000 -2.80618E+000 -6.69162E+000 7.81250E-004 2.71828E+000 -3.10721E+000 -7.29433E+000 3.90625E-004 2.71828E+000 -3.40824E+000 -7.89329E+000 1.95313E-004 2.71828E+000 -3.70927E+000 -8.44284E+000

18. Download and install Gnuplot http://www.gnuplot.info/

19. Visualization: Gnuplot Reconstruct result from MHJ - Figure 3.2 using gnuplot Gnuplot is included in Python(x,y) package! Gnuplot tutorial: http://www.duke.edu/~hpgavin/gnuplot.html http://www.duke.edu/~hpgavin/gnuplot.html Example: http://www.physics.ohio- state.edu/~ntg/780/handouts/gnuplot_quadeq_example.pdf http://www.physics.ohio- state.edu/~ntg/780/handouts/gnuplot_quadeq_example.pdf 3D Examples: http://www.physics.ohio- state.edu/~ntg/780/handouts/gnuplot_3d_example_v2.pdf

20. Using gnuplot

21. Can we explain this behavior? Computed for x=10

22. Error Analysis ε ro = Round-Off error The approximation error: Recall Eq. 3.4: The leading term in the error (j=1) is therefore:

23. The Round-Off Error (ε ro ) ε ro depends on the precision level of the chosen variables (single or double precision) Single precision Double precision If the terms are very close the difference is at the level of the round off error

24. h min = 10 -4 is therefore the step size that gives the minimal error in our case. If h>h min the round-off error term will dominate h min = 10 -4 is therefore the step size that gives the minimal error in our case. If h>h min the round-off error term will dominate

26. Let’s upgrade our visualization skills! Mayavi Included in the Python(x,y) package 2D/3D User guide: http://code.enthought.com/projects/mayavi/ docs/development/html/mayavi/index.html http://code.enthought.com/projects/mayavi/ docs/development/html/mayavi/index.html

27. A nice demo Source: http://code.enthought.com/projects/mayavi/docs/d evelopment/html/mayavi/mlab.html#id5 http://code.enthought.com/projects/mayavi/docs/d evelopment/html/mayavi/mlab.html#id5

28. The Python code

29. Mayavi environment

30. Move the object and zoom with the mouse