Numerical Methods in Biomedical Engineering

Transcription

1 Numerical Methods in Biomedical Engineering Lecture s website for updates on lecture materials: h5p://9nyurl.com/m4frahb Individual and group homework Individual and group midterm and final projects (Op?onal) computer lab every Monday, Room B03, pm

2 What are numerical methods? Techniques by which mathema?cal problems are formulated so that they can be solved with arithme?c opera?ons. They provide approxima?ons to the problems in ques?on.

3 Why study numerical methods? [Computa?onal Modeling of Endovascular Deep Brain Simula?on] hxps:// modeling- endovascular- deep- brain- simula?on Most ( > 99.9%) of real world problems in science and engineering are too complex and sophis?cated to be solved analy?cally (exactly), hence they can only be solved numerically (approximately).

4 Errors and Numerical Series

5 Errors Computers use a base-2 representation Computers cannot precisely represent certain exact base-10 numbers. Non-integer numbers, such as π = , e = , or are cumbersome and can t be expressed by a fixed number of significant figures. The discrepancy creates an error usually referred to as round-off error or rounding error

6 Errors Suppose ã is an approximation to the (nonzero) true value a, then: Absolute error Relative error Example: the value of π = is to be stored on a base-10 system that allows 7 significant figures. Chopping approximation π = Absolute error = = Rounding approximation π = Absolute error = =

7 Floa9ng- point System Fractional numbers in computers are usually represented using floating-point form: exponent man?ssa base of the number system being used Example: in a floating-point base-10 system that allows only 4 decimal places to be stored, the quantity 1/34 = would be stored as x 10-1 Allows both fractions and very large numbers to be expressed on the computer Takes up more space Takes longer time to process Source of round-off error

8 Numerical Error For numerical methods, the true value of a function is known from its analytical solution. However, in real-world applications, it is impossible to know the true value of a function a priori. Hence, the percentage relative error:

9 Maclaurin s series Let the power series for f(x) be where are constants. At

10 Substituting for in f(x) gives:

11 Condi9ons of Maclaurin s series (3) The resultant Maclaurin s series must be convergent

12 y Q y=f(x) P f(0) f(h) 0 h x Using Maclaurin s series, at some point Q in Figure above:

13 Taylor series If the y-axis and origin are moved a units to the left, the equation of the same curve relative to the new axis becomes y = f(a+x) and the function value at P is f(a). y Q y=f(a+x) P f(0) f(a) f(a+h) At point Q: 0 a h x

14 Taylor series provides a means to predict a function value at one point in terms of the function value and its derivatives at another point: where: Zero order n = order of derivative

15 Example Use zero-through third-order Taylor series expansion to predict for using a base point at. Compute the true percent relative error for each approximation. Solution The true value of the function at is, which is the value that we are going to predict/approximate. For, the Taylor series approximation is and relative error

16 For, the first derivative is, and the first order Taylor series approximation and relative error For, the second derivative is, and the second order Taylor series approximation and relative error

17 For, the third derivative is, and the third order Taylor series approximation The Taylor series expansion to the third order derivative yields an exact estimate at, hence, the remainder term is

18 Trunca9on Errors Taylor series can be used to estimate truncation errors. The notion of truncation errors usually refers to errors introduced when a more complicated mathematical expression is replaced with a more elementary formula. From the Taylor series expansion we truncate the series after the first derivative term

19 Trunca9on Errors Rearranging the equation gives us first- order approxima?on trunca?on error Using for, we get or

20 Error Propaga9on Suppose we have a function f(x) which has one dependent variable x. Assume that is an approximation of x. To assess the effect of the discrepancy between x and on the value of the function, we use The problem with evaluating is that f(x) is unknown because x is unknown. We can overcome this if: is close to x, and is continuous and differentiable We use Taylor series to compute f(x) near

21 Error Propaga9on Dropping the second- and higher-order terms and rearranging gives us or where represents an estimate of the error of the function f(x) represents an estimate of the error of x This enables us to approximate the error in f(x) given the derivative of a function and an estimate of the error in x.

22 Taylor Series for Func9ons with More than One Variable If we have a function of two independent variables x and z, the Taylor series can be written as

23 Taylor Series for Func9ons with More than One Variable If all second-order and higher terms are dropped and rearrange, we get where is the estimate of the error in x is the estimate of the error in z