advanced mathematics and numerical analysis pdf

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Numerical Analysis

“ by faith and faith alone embrace believing where we cannot prove” from In Memoriam by Alfred Lord Ten-nyson a memorial to Arthur Hallum Numerical analysis provides the foundations for a major paradigm shift in what we understand as an acceptable “answer” to a scientific or techni- cal question In classical calculus we look for answers lik

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Advanced Numerical Analysis

the solution is \\averaged out\" by the numerical methods This is a good example where mathematical analysis cannot be replaced by numerics 1 2 Simple one-step methods The (forward) Euler method [m ethode d’Euler (explicite/progressive)] is the simplest numerical method for solving an IVP Given a small step size h the idea is

What are examples of numerical analysis?
Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
What is a numerical arithmetic course?
This course offers an advanced introduction to numerical analysis, with a focus on accuracy and efficiency of numerical algorithms. Topics include sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems), floating-point arithmetic, backwards error analysis, …
Can mathematical analysis be replaced by numerics?
This is a good example where mathematical analysis cannot be replaced by numerics. The (forward) Euler method [methode d'Euler (explicite/progressive)] is the simplest numerical method for solving an IVP. Given a small step size h, the idea is to replace _y(t) in the di erential equation _y(t) = f(t; y(t)) by the forward di erence.
Is numerical analysis a mathematical subject?
As a result, numerical analysis is frequently presented as an elementary subject. As a corollary, most students miss exposure to numerical analysis as a mathemat-ical subject. We hope to provide an alternative. Several books written some decades ago addressed specifically a mathe-matical audience, e.g., [80, 84, 86].

Preface

“

1.4 AN UNSTABLE ALGORITHM

Heron’s algorithm has one drawback in that it requires division. One can imagine that a simpler algorithm might be possible such as people.cs.uchicago.edu

2.2.4 Secant method

The secant method approximates the slope by a difference method: f(xn) s = people.cs.uchicago.edu

4.1.2 Memory references

To understand the impact of memory references, we need to have some model of memory. Indeed, if memory access were very fast compared to floating-point operations (as it was in early digital computers), we could ignore its effect. But modern computer architectures have very complex memory sys-tems, with multiple levels of memory including different

✪ O(n2)

it is much more efficient to use the factorization rather than the inverse. people.cs.uchicago.edu

0 A(n−1) , (6.21)

where z = B?A?x is some (column) vector of length n people.cs.uchicago.edu

Jf(x)g(x) = Jf(x)x − f(x). (7.38)

Thus we need to differentiate the product h(x) = Jf(x)g(x) for two different functions g, so let us consider this separately. Formally, we can expect this to be of the form people.cs.uchicago.edu

7.2.3 No other methods

Unfortunately, the other methods we studied in the one-dimensional case do not generalize to multidimensions. For example, Steffensen’s method fails for two reasons. At the simplest level, it is not clear how to “divide” by the difference quotient in the vector case. But more fundamentally, the difference approximation f(x + f(x)) − f(x) would prov

Mk = I − Mn+1. (8.3)

If we are allowed to let n → ∞, then this provides a formula for the inverse of I − M: ∞ people.cs.uchicago.edu

9.2 CONJUGATE GRADIENT ITERATION

The conjugate gradient method uses a more sophisticated choice for the search directions sk. Instead of just using the current residual (sk = the direction is chosen to be orthogonal (conjugate) to previous residuals. −rk), This avoids repeating previous mistakes, or rather it avoids repeating direc-tions already traversed. people.cs.uchicago.edu

9.3 OPTIMAL APPROXIMATION OF CG

Originally, we motivated conjugate gradients as an iterative method based on minimizing a quadratic form. We have seen that CG can also be viewed as a direct method, in that it is guaranteed to reduce the residual to zero in at most n steps for an n × n system. Now we change our point of view back to our original presentation of CG: as an approxima

Polynomial Interpolation

The web site http://www.blackphoto.com/glossary/i.asp describes interpolation as “a technique used by digital cam-eras, scanners and printers to increase the size of an image in pixels by averaging the colour and brightness values of surrounding pixels.” The approximation of general functions by simple classes of functions has many applications as

11.4.1 Existence of interpolant

The existence of the generalized interpolant can be proved by constructing polynomials φi ∈ Pn such that people.cs.uchicago.edu

R = 1 2(P

∈ Pn Q) because, by the triangle inequality, kf − Rk∞,I = kf − + Q)k∞,I people.cs.uchicago.edu

12.3.2 Orthogonal polynomials

We will construct polynomials that are orthonormal: b people.cs.uchicago.edu

12.3.3 Roots of orthogonal polynomials

First, we claim that the real roots of Pn are all simple. Suppose that people.cs.uchicago.edu

13.1 INTERPOLATORY QUADRATURE

The idea behind interpolatory quadrature is to define the approximate inte-gral as the integral of an interpolant (or other approximant): Z b n Z b φi(x) dx = a n people.cs.uchicago.edu

13.1.4 Hermite quadrature

Any approximation scheme can be used to create a quadrature rule via the recipe people.cs.uchicago.edu

t x, ψk ≤ t 0,

t. However, ≡ it is easy ψk t is discontinuous at x = to see that ψk people.cs.uchicago.edu

U defined by U = R+P∞ k k=0 bkT satisfies

= f Note that we require that the operator R : but we do not require that P∞ → P∞, people.cs.uchicago.edu

D−1f(x) = Z x f(s) ds, (13.71)

but the inverse is not uniquely defined. That is, define people.cs.uchicago.edu

Exercise 13.7

Prove that the error for the trapezoidal rule on isfies people.cs.uchicago.edu

14.1.2 Quality rankings

Suppose you want to quantify connectivity of related objects based on the quality of the relationships among the objects. For example, we might want to measure personal connections so that we can target advertising to people who influence influential people. To see how this might be done, suppose we had access to all cell phone data. We can rank pe

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