application of numerical analysis in computer science

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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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Numerical Methods for Computational Science and Engineering

Numerical Methods for Computational Science and Engineering Numerical algorithms and errors Discretization errors in action Problem: want to approximate the derivative f0(x 0) of a given smooth function f(x) at the point x = x 0 Example: Let f(x) = sin(x) 1

What is numerical analysis in data science?
Numerical analysis is fundamental to data science and data analysis. It is the study of methods and algorithms that render numerical solutions, using computing machines, to mathematical problems.
What programming language is used for numerical analysis?
The most popular programming language for implementing numerical analysis methods is Fortran, a language developed in the 1950s that continues to be updated to meet changing needs. Other languages, such as C, C++, and Java, are also used for numerical analysis.
What does a numerical analyst do?
Numerical analysts are very interested in the effects of using finite precision computer arithmetic. This is especially important in numerical linear algebra, as large problems contain many rounding errors. Numerical analysts are generally interested in measuring the efficiency (or “cost”) of an algorithm.
What is the area of numerical analysis?
The area of numerical analysis is also known as numerical computation, numerical methods, numerical mathematics, computational mathematics, scientific computing or mathematical software, depending on which aspect of this broad subject is being emphasized.

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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