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CIS 541 – Numerical Methods

CIS 541 – Numerical Methods. Mathematical Preliminaries. Derivatives. Recall the limit definition of the first derivative. Partial Derivatives. Same as derivatives, keep each other dimension constant. (1,f’(t)). f(x). t. Tangents and Gradients.

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CIS 541 – Numerical Methods

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  1. CIS 541 – Numerical Methods Mathematical Preliminaries

  2. Derivatives • Recall the limit definition of the first derivative. OSU/CIS 541

  3. Partial Derivatives • Same as derivatives, keep each other dimension constant. OSU/CIS 541

  4. (1,f’(t)) f(x) t Tangents and Gradients • Recall that the slope of a curve (defined as a 1D function) at any point x, is the first derivative of the function. • That is, the linear approximation to the curve in the neighborhood of t is l(x) = b + f’(t)x y x OSU/CIS 541

  5. Tangents and Gradients • Since we also want this linear approximation to intersect the curve at the point t. l(t) = f(t) = b + f’(t)t • Or, b = f(t) - f’(t)t • We say that the line l(x)interpolates the curve f(x) at the point t. OSU/CIS 541

  6. Functions as curves • We can think of the curve shown in the previous slide as the set of all points (x,f(x)). • Then, the tangent vector at any point along the curve is OSU/CIS 541

  7. Side note on Curves • There are other ways to represent curves, rather than explicitly. • Functions are a subset of curves (x,y(x)). • Parametric equations represent the curve by the distance walked along the curve (x(t),y(t)). Circle: (cos, sin) • Implicit representations define a contour or level-set of the function: f(x,y) = c. OSU/CIS 541

  8. Tangent Planes and Gradients • In higher-dimensions, we have the same thing: • A surface is a 2D function in 3D: Surface = (x, y, f(x,y) ) • A volume or hyper-surface is a 3D function in 4D: Volume = (x, y, z, f(x,y,z) ) OSU/CIS 541

  9. Tangent Planes and Gradients • The linear approximation to the higher-dimensional function at a point (s,t), has the form: ax+by+cz+d=0, or z(x,y) = … • What is this plane? OSU/CIS 541

  10. Construction of Tangent Planes Images courtesy of TJ Murphy: http://www.math.ou.edu/~tjmurphy/Teaching/2443/TangentPlane/TangentPlane.html OSU/CIS 541

  11. Construction of Tangent Planes OSU/CIS 541

  12. Construction of Tangent Planes OSU/CIS 541

  13. Tangent Planes and Gradients • The formula for the plane is rather simple: • z(s,t) = f(s,t) - interpolates • z(s+dx,t) = f(s,t) + fx(s,t)dx = b + adx • Linear in dx • Of course, the plane does not stay close to the surface as you move away from the point (s,t). OSU/CIS 541

  14. Tangent Planes and Gradients • The normal to the plane is thus: • The 2D vector:is called the gradient of the function. • It represents the direction of maximal change. OSU/CIS 541

  15. Gradients • The gradient thus indicates the direction to walk to get down the hill the fastest. • Also used in graphics to determine illumination. OSU/CIS 541

  16. Review of Functions • Extrema of a function occur where f’(x)=0. • The second derivative determines whether the point is a minimum or maximum. • The second derivative also gives us an indication of the curvature of the curve. That is, how fast it is oscillating or turning. OSU/CIS 541

  17. The Class of Polynomials • Specific functions of the form: OSU/CIS 541

  18. The Class of Polynomials • For many polynomials, the latter coefficients are zero. For example: p(x) = 3+x2+5x3 OSU/CIS 541

  19. Taylor’s Series • For a function, f(x), about a point c. • I.E. A polynomial OSU/CIS 541

  20. Taylor’s Theorem • Taylor’s Theorem allows us to truncate this infinite series: OSU/CIS 541

  21. Taylor’s Theorem • Some things to note: • (x-c)(n+1) quickly approaches zero if |x-c|<<1 • (x-c)(n+1) increases quickly if |x-c|>>1 • Higher-order derivatives may get smaller (for smooth functions). OSU/CIS 541

  22. Higher Derivatives • What is the 100th derivative of sin(x)? • What is the 100th derivative of sin(3x)? • Compare 3100 to 100! • What is the 100th derivative of sin(1000x)? OSU/CIS 541

  23. Taylor’s Theorem • Hence, for points near c we can just drop the error term and we have a good polynomial approximation to the function (again, for points near c). • Consider the case where (x-c)=0.5 • For n=4, this leads to an error term around 2.6*10-4 f() • Do this for other values of n. • Do this for the case (x-c) = 0.1 OSU/CIS 541

  24. Some Common Derivatives OSU/CIS 541

  25. Some Resulting Series • About c=0 OSU/CIS 541

  26. Some Resulting Series • About c=0 OSU/CIS 541

  27. Book’s Introduction Example • Eight terms for first series not even yielding a single significant digit. • Only four for second serieswith foursignificantdigits. OSU/CIS 541

  28. Mean-Value Theorem • Special case of Taylor’s Theorem, where n=0, x=b. • Assumes f(x) is continuous and its first derivative exists everywhere within (a,b). OSU/CIS 541

  29. Mean-Value Theorem • So what!?!?! What does this mean? • Function can not jump away from current value faster than the derivative will allow. f(x) a secant b  OSU/CIS 541

  30. Rolles Theorem • If a and b are roots (f(a)=f(b)=0) of a continuous function f(x), which is not everywhere equal to zero, then f’(t)=0 for some point t in (a,b). • I.e., What goes up, must come down. f(x) f’(t)=0 a t b OSU/CIS 541

  31. Caveat • For Taylor’s Series and Taylor’s Theorem to hold, the function and its derivatives must exist within the range you are trying to use it. • That is, the function does not go to infinity, or have a discontinuity (implies f’(x) does not exist), … OSU/CIS 541

  32. Implementing a Fast sin() const int Max_Iters = 100,000,000; float x = -0.1; float delta = 0.2 / Max_Iters; float Reimann_sum = 0.0; for (int i=0; i<Max_Iters; i++) { Reimann_Sum += sinf(x); x+=delta; } Printf(“Integral of sin(x) from –0.1->0.1 equals: %f\n”, Reimann_Sum*delta ); OSU/CIS 541

  33. Implementing a Fast sin() const int Max_Iters = 100,000,000; float x = -0.1; float delta = 0.2 / Max_Iters; float Reimann_sum = 0.0; for (int i=0; i<Max_Iters; i++) { Reimann_Sum += my_sin(x); //my own sine func x+=delta; } Printf(“Integral of sin(x) from –0.1->0.1 equals: %f\n”, Reimann_Sum*delta ); OSU/CIS 541

  34. Version 1.0 my_sin( const float x ) { float x2 = x*x; float x3 = x*x2; return (x – x3/6.0 + x2*x3/120.0 ); } OSU/CIS 541

  35. Version 2.0 – Horner’s Rule Static const float fac3inv = 1.0 / 6.0f; Static const float fac5inv = 1.0 / 120.0f; my_sin( const float x ) { float x2 = x*x; return x*(1.0 – x2*(fac3inv - x2*fac5inv)); } OSU/CIS 541

  36. Version 3.0 – Inline code const int Max_Iters = 100,000,000; float x = -0.1; float delta = 0.2 / Max_Iters; float Reimann_sum = 0.0; for (i=0; i<Max_Iters; i++) { x2 = x*x; Reimann_Sum += x*(1.0–x2*(fac3inv-x2*fac5inv); x+=delta; } Printf(“Integral of sin(x) from –0.1->0.1 equals: %f\n”, Reimann_Sum*delta ); OSU/CIS 541

  37. Timings • Pentium III, 600MHz machine OSU/CIS 541

  38. Observations • Is the result correct? • Why did we gain some accuracy with version 2.0? • Is (–0.1,0.1) a fair range to consider? • Is the original sinf() function optimized? • How did we achieve our speed-ups? • We will re-examine this after Lab1. Ask these question for Lab1 !!! OSU/CIS 541

  39. Homework • Read Chapters 1 and 2 for next class. • Start working on Lab 1 and Homework 1. OSU/CIS 541

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