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, 56.30pm
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Numerical*Methods*in** BiomedicalEngineering*people.bu.edu/andasari/courses/Fall2014/LectureNotes/Week1_Lecture1.pdfnumber of significant figures. ... Taylor*series* If the y-axis
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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, 5-‐6.30pm
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.
Why study numerical methods?
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).
[Computa?onal Modeling of Endovascular Deep Brain Simula?on] hXps://www.msi.umn.edu/content/computa?onal-‐modeling-‐endovascular-‐deep-‐brain-‐simula?on
Errors and Numerical Series
Errors • Computers use a base-2 representation
1 0 1 0 1 1 0 1
• Computers cannot precisely represent certain exact base-10 numbers. Non-integer numbers, such as π = 3.1415926535…, e = 2.718281…, 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
Errors Suppose ã is an approximation to the (nonzero) true value a, then:
Absolute error
Relative error
Example: the value of π = 3.1415926535… is to be stored on a base-10 system that allows 7 significant figures. Chopping approximation π = 3.141592 Absolute error = |3.1415926535 – 3.141592| = 0.0000006535 Rounding approximation π = 3.141593 Absolute error = |3.1415926535 – 3.141593| = 0.0000003465
Floa9ng-‐point System Fractional numbers in computers are usually represented using floating-point form:
man?ssa base of the number system being used
exponent
Example: in a floating-point base-10 system that allows only 4 decimal places to be stored, the quantity 1/34 = 0.029411765 would be stored as 0.2941 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
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:
Maclaurin’s series
Let the power series for f(x) be
where are constants.
At
Substituting for in f(x) gives:
Condi9ons of Maclaurin’s series
(3) The resultant Maclaurin’s series must be convergent
0
f(0) f(h)
x
y=f(x)
y
h
P
Q
Using Maclaurin’s series, at some point Q in Figure above:
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).
0
f(0)
f(a+h)
x
y=f(a+x) y
a
P
Q
At point Q:
f(a)
h
Zero order
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:
n = order of derivative
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
For , the first derivative is , and the first order Taylor series approximation
For , the second derivative is , and the second order Taylor series approximation
and relative error
and relative error
For , the third derivative is , and the third order Taylor series approximation
hence, the remainder term is
The Taylor series expansion to the third order derivative yields an exact estimate at ,
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
Trunca9on Errors
Using
for , we get
or
Rearranging the equation gives us
trunca?on error first-‐order approxima?on
Error Propaga9on
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
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
Error Propaga9on
where
or
Dropping the second- and higher-order terms and rearranging gives us
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.
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
Taylor Series for Func9ons with More than One Variable
If all second-order and higher terms are dropped and rearrange, we get