2 differentiation mod.ppt [compatibility mode]

Introduction toNumerical Analysis for Engineers

Differentiation

Function Evaluation (1)

Problem: Compute f(x)

Easy for well known functionsFor example, f (x) = 1, f (x) = x2

Function Evaluation (2) Suppose f (x) is a black box

Difficult and/or expensive to compute For example: a complex function, an experiment, a procedure

Limited information about the function Suppose we know information about the function at a particular

domain location, x = c f (c), f ’(c), f ’’(c), f ’’’(c), …

x f (x)f

Taylor’s Series Evaluate f (x) about a point c, called the point of

expansion

Observations: An infinite series (x – c) is the distance from the point of expansion F (x) is a polynomial!!!

Some Common Derivatives

October 21, 2013 6Introduction Numerical Analysis for Engineer

Numerical Differentiation

Analytically, Calculus defines for us:

Geometrically: The tangent (line in 1D)

x


Numerical Differentiation How do we program this?

The limit as h→ 0 … cannot set h = 0, so must approximate! What value of h is appropriate?

Choose a non-zero h Geometrically, rise/run or the slope of the secant curve

x x + h

Tangent Line

Secant Line


Computing Errors

Analyze

Any precision problems???

Take 1/h As h gets smaller, this value becomes larger … good As h gets larger, this value gets smaller

… Truncation Error


Round Off Error

Analyze

As h gets smaller, f (x) and f (x + h) get closer

x + hx x + h


Round Off Error

Thus, f (x + h) – f (x) → 0 Loss of significance Easily get a random round-off bit as the most

significant digit in the subtraction

So, dividing by h leads to a very wrong answerfor f ’(x)


Error Tradeoff As h gets smaller, truncation error decreases, but round off error increases

The trade off is a point of diminishing returns Always think and test!

Log error

Log step size

Truncation error

Round off error

Total errorPoint of

diminishingreturns



This formula favors (or biasestowards) the right-hand side of the curve

Why not use the left? (f (x) – f(x - h))/h

x x + h

Secant Line

x - h


Error Analysis … Taylor’s Theorem

Forward Differences Take Taylor’s to three terms … let’s take a look at

the error term

Theoreticallyspeaking

Contoh


Gunakan forward difference, dengan nilai h seperti dibawah ini, untuk menghitung turunan pertama dengannumerik dari cos(x) pada x = π/3.(a) h = 0.1(b) h = 0.01(c) h = 0.001(d) h = 0.0001Hitung sampai 8 decimal.

Contoh


Solusi eksak



Backward Differences


Contoh


Gunakan backward difference, dengan nilai h seperti dibawah ini, untuk menghitung turunan pertama dengannumerik dari cos(x) pada x = π/3.(a) h = 0.1(b) h = 0.01(c) h = 0.001(d) h = 0.0001Hitung sampai 8 decimal.



Linear Convergence


Numerical Differentiation You should be asking yourself … Can we do better?

Average both sides together???

x x + hx - h

f (x + h) – f (x – h)

2h



Central Difference

Is this any better?

Forward difference Backward difference


Central Differences

This formula does not seem very good Does not follow the calculus formula We don’t even evaluate at x (our location of interest)

It takes the slope of the secant with width 2h

x x + hx - h

Secant Line



OK, let’s stop guessing … Taylor’s will tell us

Solve for this

Four terms


Central Differences

Central differences has much better convergence As h2 → 0, the error term has quadratic convergence


Contoh


Gunakan central difference, dengan nilai h seperti di bawahini, untuk menghitung turunan pertama dengan numerikdari cos(x) pada x = π/3.(a) h = 0.1(b) h = 0.01(c) h = 0.001(d) h = 0.0001Hitung sampai 8 decimal.

Contoh


Solusi eksak


Let f(x) = cosh(x) and a = 2. Let h = 0.01 andapproximate f’ (a), f’’(a), using forward, backwardand central differences.Work to 8 decimal places and compare youranswers with the exact result,

Five Point Formula



Second Derivatives

What if we need the second derivative?

Any guesses?


Second Derivatives

Add the terms Cancel the odd terms and double up the even ones


Second Derivatives

Isolating the second derivative term yields:

With an error term of:

Third Derivative


Fourth Derivative


Contoh



Partial Derivatives

Nothing really special about partial derivatives Hold the other dimensions constant

Partial Derivatives


2 differentiation mod.ppt [compatibility mode]

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