Post on 18-Jan-2016
Today’s class
• Numerical differentiation• Roots of equation• Bracketing methods
Numerical Methods, Lecture 4 1
Prof. Jinbo Bi CSE, UConn
• Finite divided difference
• First forward difference
• First backward difference
Numerical Differentiation
Numerical Methods, Lecture 4 2
Prof. Jinbo Bi CSE, UConn
• Centered difference approximation
• Subtract the two equations
Numerical Differentiation
Numerical Methods, Lecture 3 3
Prof. Jinbo Bi CSE, UConn
• First forward difference
Numerical Differentiation
Numerical Methods, Lecture 4 4
Prof. Jinbo Bi CSE, UConn
• First backward difference
Numerical Differentiation
Numerical Methods, Lecture 4 5
Prof. Jinbo Bi CSE, UConn
• Centered difference
Numerical Differentiation
Numerical Methods, Lecture 4 6
Prof. Jinbo Bi CSE, UConn
• What is the effect of error in one calculation propagating to subsequent calculations?
• Example:• Multiplying sin x with cos x
• Single variable functions
Error Propagation
Numerical Methods, Lecture 4 7
Prof. Jinbo Bi CSE, UConn
• Use Taylor series
Error Propagation
Numerical Methods, Lecture 4 8
Prof. Jinbo Bi CSE, UConn
Error Propagation
Numerical Methods, Lecture 4 9
Prof. Jinbo Bi CSE, UConn
• Multivariable functions
Error Propagation
Numerical Methods, Lecture 4 10
Prof. Jinbo Bi CSE, UConn
• Condition of a problem is a measure of its sensitivity to changes in input values
• The condition number is defined as the ratio of the relative function error to the relative value error
Numerical stability
Numerical Methods, Lecture 4 11
Prof. Jinbo Bi CSE, UConn
• Condition number < 1 indicates a well-conditioned function – i.e. changes in the input are attenuated
• Condition number > 1 indicates a ill-conditioned function – i.e. changes in the input are amplified
Numerical stability
Numerical Methods, Lecture 4 12
Prof. Jinbo Bi CSE, UConn
Roots of equation• Given a function f(x), the roots are those
values of x that satisfy the relation f(x) = 0• Example
• From the quadratic formula, the roots are:
Numerical Methods, Lecture 4 13
Prof. Jinbo Bi CSE, UConn
Roots of Equations• The need to solve for roots show up in
many engineering problems• Also, can be used to find solutions to
implicit variables
Numerical Methods, Lecture 4 14
Prof. Jinbo Bi CSE, UConn
• Find a value of R such that current is 5A at t = 1s
Example
Numerical Methods, Lecture 4 15
Prof. Jinbo Bi CSE, UConn
Example
• It is not possible to isolate R to the left side and thus solve for R
• R is know as an implicit variable• Rewrite the function as a function of R
set to 0
Numerical Methods, Lecture 4 16
Prof. Jinbo Bi CSE, UConn
• Still need a method to solve for this root• Other examples of difficult to solve roots
Roots of equations
Numerical Methods, Lecture 4 17
Prof. Jinbo Bi CSE, UConn
• Non-computer methods• Graphical methods
Roots of equations
Numerical Methods, Lecture 4 18
Prof. Jinbo Bi CSE, UConn
• Not exact• Can give you a rough estimate of the root, • Can give you insights on the number of roots
and shape of the curve• Can use the rough estimate in more precise
numerical methods
Graphical methods
Numerical Methods, Lecture 4 19
Prof. Jinbo Bi CSE, UConn
• Use to get an initial estimate of the root and also to find out how many roots there are
Graphical methods
Numerical Methods, Lecture 4 20
Prof. Jinbo Bi CSE, UConn
Graphical methods
Numerical Methods, Lecture 4 21
Prof. Jinbo Bi CSE, UConn
Graphical methods
Numerical Methods, Lecture 4 22
Prof. Jinbo Bi CSE, UConn
Graphical methods
Numerical Methods, Lecture 4 23
Prof. Jinbo Bi CSE, UConn
• Non-computer/numerical method• Exhaustive search method
• To find the root in the interval [a,b], start at x=a and check if f(a) = 0, then try f(a+Δ), f(a+2Δ), and so on, until we get f(x) sufficiently close to 0
• If the step value Δ is sufficiently small we can obtain an accurate result but this could take an extremely long time. For example, if the interval is [0,10] and the step size is Δ = 0.001, it will take on average 10,000 guesses
• In addition to the inefficiency of this approach, if f(x) is a steep function, this approach may not produce an accurate results
Roots of equation
Numerical Methods, Lecture 4 24
Prof. Jinbo Bi CSE, UConn
• Example• Find the root of the function • Actual root is at x=1.0001• With an interval of [0.9, 1.1] and a step size of
Δ = 0.001. The exhaustive search method will test f(1.000) = -0.01 and f(1.001) = 0.086, neither of which are that close to f(x) = 0
Exhaustive search
Numerical Methods, Lecture 4 25
Prof. Jinbo Bi CSE, UConn
• More systematic methods are required• Bracketing methods
• Open methods
Roots of equations
Numerical Methods, Lecture 4 26
Prof. Jinbo Bi CSE, UConn
• Locate an interval where sign changes• Divide interval into smaller subintervals
which are then searched for sign changes• Keep repeating until root is found with
sufficient confidence
Incremental search methods
Numerical Methods, Lecture 4 27
Prof. Jinbo Bi CSE, UConn
• Also called:• Binary chopping
• Interval halving
• An incremental search method where the interval is cut in half
Bisection method
Numerical Methods, Lecture 4 28
Prof. Jinbo Bi CSE, UConn
• Step 1:• Choose lower xl and upper xu such that the
function changes sign over that range – i.e. f(x l) and f(xu) are different signs – or f(xl) f(xu) < 0
• Step 2:• Estimate root to be xr=(xl+xu)/2
Bisection method
Numerical Methods, Lecture 4 29
Prof. Jinbo Bi CSE, UConn
• Step 3:• Determine in which subinterval the root lies
• If f(xr)0 is within acceptable tolerance, stop and root equals xr
• If f(xl) f(xr) < 0, then root is in lower subinterval. Set xu = xr, and return to step 2
• If f(xl) f(xr) > 0, then root is in upper subinterval. Set xl = xr, and return to step 2
Bisection method
Numerical Methods, Lecture 4 30
Prof. Jinbo Bi CSE, UConn
• Termination criteria• Use approximate relative error calculation to
determine when to stop
• In general, a is larger than t
Bisection method
Numerical Methods, Lecture 4 31
Prof. Jinbo Bi CSE, UConn
• Example:
• Use range of [202:204]
• Root is in upper subinterval
Bisection method
Numerical Methods, Lecture 4 32
Prof. Jinbo Bi CSE, UConn
Bisection method
• Use range of [203:204]
• Root is in lower subinterval
–0.0034
Numerical Methods, Lecture 4 33
Prof. Jinbo Bi CSE, UConn
• Use range of [203:203.5]
• Root is in upper subinterval
Bisection method
Numerical Methods, Lecture 4 34
Prof. Jinbo Bi CSE, UConn
• The approximate error is upper bound estimate of the true error
• When the root is near one of the ends of the interval, the approximate error is fairly close to the actual true error
• Error is fairly well-contained
Error estimates
Numerical Methods, Lecture 4 35
Prof. Jinbo Bi CSE, UConn
• You always know that the true root is within Δx/2 of your estimate
Error estimates
Numerical Methods, Lecture 4 36
Prof. Jinbo Bi CSE, UConn
Bisection method
Numerical Methods, Lecture 4 37
Prof. Jinbo Bi CSE, UConn
• You can calculate an error estimate based just on the initial guesses
• You can also make estimates on the error on future iterations
• Superscripts indicates the iteration number
Bisection method
Numerical Methods, Lecture 4 38
Prof. Jinbo Bi CSE, UConn
• Each subsequent iteration cuts the approximate error in half
• This, allows to determine a priori exactly how many iterations are needed to arrive at the desired error
Bisection method
Numerical Methods, Lecture 4 39
Prof. Jinbo Bi CSE, UConn
• The false position method works in a similar fashion to the bisection method
• Start with an initial interval [a,b] where f(a) and f(b) have opposite signs, which is the same as the bisection
• Instead of choosing the initial guess xr as the midpoint of the interval, we join the point {a,f(a)} and {b,f(b)} with a straight line and choose xr as the point where that straight line crosses the x-axis.
False Position Method
Numerical Methods, Lecture 4 40
Prof. Jinbo Bi CSE, UConn
False Position MethodCopyright © The McGraw-Hil l Companies, Inc. Permission required for reproduction or d isplay.Fig 5.12
Numerical Methods, Lecture 4 41
Prof. Jinbo Bi CSE, UConn
• Algorithm is the same as bisection method with the same three steps
False Position Method
Numerical Methods, Lecture 4 42
Prof. Jinbo Bi CSE, UConn
• Step 1:• Choose lower xl and upper xu such that the
function changes sign over that range - i.e. f(xl) and f(xu) are different signs - or f(xl) f(xu) < 0
• Step 2:• Estimate new root to be
False Position Method
Numerical Methods, Lecture 4 43
Prof. Jinbo Bi CSE, UConn
• Step 3:• Determine in which subinterval the root lies
• If f(xr) 0 is within acceptable tolerance, stop and root equals xr
• If f(xl) f(xr) < 0, then root is in lower subinterval. Set xu = xr, and return to step 2
• If f(xl) f(xr) > 0, then root is in upper subinterval. Set xl = xr, and return to step 2
False Position Method
Numerical Methods, Lecture 4 44
Prof. Jinbo Bi CSE, UConn
• Roots of equations• Open methods• Read chapters 5 and 6• HW2, due 9/17
• Chapra & Canale• 6th edition 3.5, 3.7, 3.13, 4.5, 4.6, 4.12 (b) and (d),
and 4.16
Next class
Numerical Methods, Lecture 4 45
Prof. Jinbo Bi CSE, UConn