22 - 3/3/2000AME 150L1 Numerical Integration. 22 - 3/3/2000AME 150L2 What You Know 125 Calculus I (4) Limits; continuity, derivatives and applications;

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Presentation transcript:

22 - 3/3/2000AME 150L1 Numerical Integration

22 - 3/3/2000AME 150L2 What You Know 125 Calculus I (4) Limits; continuity, derivatives and applications; antiderivatives; the fundamental theorem of calculus; exponential and logarithmic functions. Prerequisite: MATH 108 or placement exam. 126 Calculus II (4) A continuation of MATH 125: trigonometric functions; applications of integration; techniques of integration; indeterminate forms; infinite series; Taylor series; polar coordinates. Prerequisite: MATH 125.

22 - 3/3/2000AME 150L3 Numerical Differentiation Differentiation comes from the limit of a difference However, if y(x) has a small amount of noise, differentiation enhances the noise Every differentiation involves subtraction of two numbers close to each each

22 - 3/3/2000AME 150L4 Noisy Functions y(x)=f(x)+e(x) or sin(x)+  (x) y x

22 - 3/3/2000AME 150L5 Finite Differences Taylor's Theorem –Functions are continuous –All derivatives exist at point x=x 0 –Can expand function around point x=x 0

22 - 3/3/2000AME 150L6 Truncation error (if stop at  f )

22 - 3/3/2000AME 150L7 Finite Differences

22 - 3/3/2000AME 150L8 Numerical Integration Integration is same as calculation of area Integration reduces noise Integration adds two numbers that are close together (it does not subtract them)

22 - 3/3/2000AME 150L9 Trapezoidal Rule y x y=f(x) ba

22 - 3/3/2000AME 150L10 Area Formed by a Number of Small Trapezoids x1=ax1=ax2x2 X3X3 y1y1 y2y2 y3y3 h=x j+1 -x j N=(b-a)/h X4X4 X5=bX5=b

22 - 3/3/2000AME 150L11 Trapezoidal Rule (concluded) I  I h =(h/2)(y 0 +2y 1 +2y y n-2 +2y n-1 +y n ) where h=x j+1 -x j Truncation error can be computed to be |error T |  h 2 (b-a) max |y''(  )|/12

22 - 3/3/2000AME 150L12 Simpson´s Rule Trapezoidal rule passes straight line through 2 adjacent points Can pass a parabola through 3 adjacent points (and calculate area) I  (y 0 +y n )h/3 + (y 1 +y 3 +…+y n-3 +y n-1 )4h/3 +(y 2 +y 4 + +y n-4 +y n-2 )2h/3 I  h/3(  ends +4  odds +2  evens ) - h 4 f iv (x)(b-a)/180

22 - 3/3/2000AME 150L13