Numerical
Differentiation
Finite Difference Formuia: comes from a sim
ple approximation based on computing the sIQpe
between two points (t§f(t), (t+At,f(tiAt),
that is the secant line
f(t + At) - f(t)
At Taking the limit of the secant line as At + O
we
Numerical Analysis II
Dr Abigail Wacher
Let us use a decimal floating point system F0 with 4 digits and an exponent between -9 and 9.
b = 10, t = 4, u =K = 9 L (Associativity) 4 4 Take a = 1.000$100 , b = 3.000$10K , c = 4.000$10K
a C b = 1.000$100
PDE for Finance, Spring 2003 Robert V. Kohn Professor of Mathematics Courant Institute, New York University
This course assumed a working familiarity with stochastic differential equations (e.g. Ito's lemma). Here are some notes that review this material:
Practical 0: Maple Refresher
Start by launching Maple: Start Programs Programming Languages Maple 12 Maple 12 While youre at it, drag the icon to your desktop for later use. Before you start, the following setup will make it easier to see what you are doi
Homework 1 due Thursday October 14 at 12:00 at lecture
Prove that the equation x3 + x + 1 = 0 has one and only one real root, p, and show that it lies between 1 and 0. Use a bisection method to nd p correct to three decimal places. How many steps of the
Homework 2 due Thursday October 21 before 13:30 in corresponding group folder on door of oce CM110
Let a > 0. Could the iterative formula pn+1 = a/pn be used to solve x2 = a, i.e. to nd the square root of a? Give reasons for your answer. Let g (x) := a/x
Homework 3 due Thursday October 28 before 13:30 in corresponding group folder on door of oce CM110
Consider the sequence cfw_pn generated by p0 = 0 , pn+1 = (p3 + 2)/6, n n = 0, 1, 2, .
Show that for all n, 1 pn 0 and that the sequence converges to a li
Homework 4 due Thursday November 4 before 13:30 in corresponding group folder on door of oce CM110
Prove that the sequence generated by the iterative formula pn+1 = 1 sin pn converges to the real root of the equation sin x + x 1 = 0 for every real starti
Homework 5 due Thursday November 11 before 13:30 in corresponding group folder on door of oce CM110
1. (a) The equation x3 + x 1 = 0 has one real root near to 0.7. Discuss briey, and without actually implementing the iterative method, the convergence of i
Homework 6 due Thursday November 18 before 13:30 in corresponding group folder on door of oce CM110
1. Prove that ( 10 + 3)5 = 1405 10 4443.
Estimate the errors involved in evaluating each of these expressions, given an ap proximation for 10 with a small
Homework 7 due Thursday November 25 before 13:30 in corresponding group folder on door of oce CM110
1. Show that the table x 1.0 1.2 1.4 1.6 1.8 2.0 f (x) 2.25 1.73 0.97 0.07 1.43 3.15 is consistent with the assumption that f (x) is a cubic polynomial in
Homework 8 due Thursday December 2 before 13:30 in corresponding group folder on door of oce CM110
1. Convince yourself that the polynomial of degree n which agrees with a function f at distinct nodes x0 , x1 , . . . , xn may be written as pn (x) = f (xn
Homework 9 due Thursday December 9 before 13:30 in corresponding group folder on door of oce CM110
1. Find the Hermite interpolating cubic for 1/(1 + x2 ), based on the values of this function and its derivative at 1 and 1. We use the Hermite interpolatio
Homework 10 due Thursday December 16 before 13:30 in corresponding group folder on door of oce CM110
1. By Taylor expansion, nd the leading term in the truncation error of the approximation f (x) [4f (x + h) 3f (x) f (x + 2h)]/2h. If there is a possible e
Tutorial week 2
Let g (x) = 2 1 + x 2, p0 [a, b] = [0, 3] and pn+1 = g (pn ). Show that cfw_pn is a strictly decreasing sequence and bounded by 0 deduce that the iteration converges. Use the Convergence of a Monotonic Sequence Theorem. We know g (0) = 0
Tutorial week 4
1. (a) Let f (x) = ex 4x. Using the Intermediate Value Theorem prove that the equation f (x) = 0 has a unique solution, p, in the interval (0, 1). (b) One rearrangement of this equation is x = g (x) := 0.25 ex . Show that the iterative met
Tutorial week 10
1. A dierentiable function takes the values f (0.1) = 1.629, f (0) = 1.312 and f (0.1) = 1.043.
Using the central and backward dierence formulae f (x2 ) f (x0 ) x2 x0 ANSWER: Using the central dierence formula with x2 = 0.1 and x0 = 0.1 f
Numerical Analysis II
Dr Abigail Wacher
Order of convergence (as in online notes) Definition Let pn converge to p and suppose pn sp c n The method converges with order if pn C 1 K p
limn /N =l pn K p a If a = 1 and l s 0 then convergence is linear If a
Durham University
EXAMINATION PAPER
date exam code
May/June 2009
MATH2051/01
description
Numerical Analysis II
Time allowed: 3 hours Examination material provided: None Instructions: Credit will be given for the best FOUR answers from Section A and the be
Numerical Analysis II
Dr Abigail Wacher
xn+1 = axn +b O f:= x->x; g:= x->2*x-2; plot([f,g],0.5,colour = [red, blue]); f := x/x g := x/2 x K 2
8
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0
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K 2
If the sequence converges it's limit is the fixed point of the function g(x)=ax+b. This i
Numerical Analysis II
Dr Abigail Wacher
Aside Continuity Recall that f:-> is continuous if > 0 > 0 s.t. |x-y|< => |f(x)-f(y)|< But some functions are "more continuous" than others, so we need a more refined definition. Definitions: Let I =d , then f:I->I
Numerical Analysis II
Dr Abigail Wacher
Convergence Speed & Acceleration Convergence of an iteration sequence can be very slow, i.e. many iterations may be required to reach the desired accuracy. When possible we would like to accelerate. We begin by unde
Numerical Analysis II
Dr Abigail Wacher
Newton's Method fx g x = xK f' x f x f ' x Since g ' x = f' x 2 g ' p = 0 as long as f ' p s0 so care must be taken if f has a double root or higher multiplicity root. Provided the starting value is "close" to p and
Numerical Analysis II
Dr Abigail Wacher
Errors Errors can be introducted in several ways. We can have experimental errors (such as measurement errores), machine/computer errors, mathematical approximations. Mathematical approximations, and computer errors
Maple can handle a very large number of digits, the maximum number of digits in our computer. O Maple_floats(MAX_DIGITS); 38654705646 The result of the function O D_H=evalhf(Digits); (1) (1)
D_H = 15.
(2)
is the number of decimal digits that the correspon
Numerical Analysis II
Dr Abigail Wacher
3. Polynomial Interpolation
When presented with n C 1 data points xi , f xi i = 0, 1,., n we may want to draw a curve to go through these points in order to obtain some information, that is by interpolation or extra
The lagrange form is a direct way to obtain the unique polynomial interpolating a set of data points which goes through each n C 1 data point, with a polynomail of order n. Given n C 1 data points x0 , y0 , x1 , y1 $.$ xn , yn where no two xj are the same
Numerical Analysis II
Dr Abigail Wacher
Two drawbacks of Lagrange formula for obtaining the interpolating polynomial. 1) To evaluate the polynomial there is a lot of arithmetic. 2) If a new node is added to the date say xn C 1 and we wish to include it in