## 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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