Numerical Analysis Home

## Introduction

Sometimes we cannot solve a problem analytically. For example,

Find the root $$x^*$$ of $$f(x)=e^x-x-2$$ on the interval $$[0,2]$$.

Also we do not have a general analytic formula or technique to find roots of a polynomial of degree 5 or more (See Galois Theory). We solve these kinds of problem numerically:

• Construct a sequence $$\{x_n\}$$ that converges to $$x^*$$, i.e., $$\displaystyle\lim_{n\to \infty} x_n=x^*$$.

• Approximate $$x^*$$ by finding $$x_k$$ for some $$k$$ for which $$f(x_k)=e^{x_k}-x_k-2 \approx 0$$.

Numerical Analysis includes study of the following:

• Construct a sequence $$\{x_n\}$$ that converges to the solution (Iteration formula)

• Determine how fast $$\{x_n\}$$ converges to the solution (Rate of convergence)

• Find bounds of error committed at a certain iteration $$x_n$$ (Error analysis)

We will cover numerical methods for the following topics:

• Root finding

• Interpolation

• Differentiation and integration

• Differential equations

• Liner algebra

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