CIS 541 – Numerica Methods Department of Computer 顺541–数值方法计算机系.pptVIP

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2018-06-19 发布于福建
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CIS 541 – Numerica Methods Department of Computer 顺541–数值方法计算机系.ppt

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CIS 541 – Numerica Methods Department of Computer 顺541–数值方法计算机系

* OSU/CIS 541 * Example We can look at the (theoretical) error term on this example. Taking the derivative: 2-144 Round-off error * OSU/CIS 541 * Second Derivatives What if we need the second derivative? Any guesses? * OSU/CIS 541 * Second Derivatives Let’s cancel out the odd derivatives and double up the even ones: Implies adding the terms together. * OSU/CIS 541 * Second Derivatives Isolating the second derivative term yields: With an error term of: * OSU/CIS 541 * Partial Derivatives Remember: Nothing special about partial derivatives: * OSU/CIS 541 * Calculating the Gradient For lab 2, you need to calculate the gradient. Just use central differences for each partial derivative. Remember to normalize it (divide by its length). OSU/CIS 541 OSU/CIS 541 CSE 541 - Differentiation Roger Crawfis * OSU/CIS 541 * Numerical Differentiation The mathematical definition: Can also be thought of as the tangent line. x x+h * OSU/CIS 541 * Numerical Differentiation We can not calculate the limit as h goes to zero, so we need to approximate it. Apply directly for a non-zero h leads to the slope of the secant curve. x x+h * OSU/CIS 541 * Numerical Differentiation This is called Forward Differences and can be derived using Taylor’s Series: Theoretically speaking * OSU/CIS 541 * Truncation Errors Let f(x) = a+e, and f(x+h) = a+f. Then, as h approaches zero, ea and fa. With limited precision on our computer, our representation of f(x) ? a ? f(x+h). We can easily get a random round-off bit as the most significant digit in the subtraction. Dividing by h, leads to a very wrong answer for f’(x). * OSU/CIS 541 * Error Tradeoff Using a smaller step size reduces truncation error. However, it increases the round-off error. Trade off/diminishing returns occurs: Always think and test! Log error Log step size Truncation error Round off error Total error Point of diminishing returns * OSU/CIS 541 * Numerical Differentiation This formula favors (or biases towards) the right-hand sid