Today’s class Numerical differentiation Roots of equation Bracketing methods Numerical Methods,...

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



• Centered difference approximation

• Subtract the two equations




• First forward difference




• First backward difference




• Centered difference




• 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



• Use Taylor series

Error Propagation



Error Propagation



• Multivariable functions

Error Propagation



• 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



• 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



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:



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



• Find a value of R such that current is 5A at t = 1s

Example



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



• Still need a method to solve for this root• Other examples of difficult to solve roots

Roots of equations



• Non-computer methods• Graphical methods

Roots of equations



• 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



• Use to get an initial estimate of the root and also to find out how many roots there are

Graphical methods



Graphical methods



• 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



• 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



• More systematic methods are required• Bracketing methods

• Open methods

Roots of equations



• 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



• Also called:• Binary chopping

• Interval halving

• An incremental search method where the interval is cut in half

Bisection method



• 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



• 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



• Termination criteria• Use approximate relative error calculation to

determine when to stop

• In general, a is larger than t

Bisection method



• Example:

• Use range of [202:204]

• Root is in upper subinterval

Bisection method



Bisection method

• Use range of [203:204]

• Root is in lower subinterval

–0.0034



• Use range of [203:203.5]

• Root is in upper subinterval

Bisection method



• 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



• You always know that the true root is within Δx/2 of your estimate

Error estimates



Bisection method



• 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



• 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



• 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



• Algorithm is the same as bisection method with the same three steps




• 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




• 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




• 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

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Today’s class Numerical differentiation Roots of equation Bracketing methods Numerical Methods,...

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