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CIS 541 – Numerical Methods

This tutorial covers the mathematical preliminaries of derivatives, partial derivatives, tangents, gradients, and tangent planes. It explains how to find the gradients of functions and use them to determine the direction of maximal change. The tutorial also introduces Taylor's series and Taylor's theorem for polynomial approximations.

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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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