# Numerical Computation Class 11 Mathematics Solutions | Exercise – 19

__Introduction__

__Introduction__

Matching of one set of objects with another (pebbles with sheep by early man) gave rise to what is known as counting the earliest form of computing. Computing usually involves number; i.e., computing is mostly numeric or numeral. But modern computer performs non- numeric computation also.

Computing may be done mentally, manually, mechanically, analytically and electronically. In the early days, computation was done with the help of sets of objects. In Vedic times, computations or calculations were done mentally. Computation in ancient Egypt, China and Babylonia were carried out using various kinds of hieroglyphs and other number symbols.

### Characteristics of Numerical Computing

Computation, in the early days, most probably began with some kind of measurement. Most measurements are quantified numerically or expressed in terms of some number (called value). In some cases, such a value becomes exact or accurate. Estimates are made when exact values could not be found.

Errors do appear when estimates are made. In many occasions, we are required to arrive at the exact value or an estimated or desired value as rapidly as possible or expected value. Unfortunately, certain computation goes wild (become unacceptable or unstable) after a certain stage or before we arrive at a desired or expected value or limit.

Most numerical computations depend upon the method of computation which we apply. A computational method deals commonly with procedures for solving problems with a computing device or computer. Any such method is judged on the basis of the following characteristics:

- Accuracy
- Rate of convergence
- Numerical stability, and
- Efficiency

### Exercise – 19.1

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We already have discussed a lot about Numerical Computation in our previous article. I’m talking about the article where I have provided you the complete notes of exercise 19.1. In that article I discussed about characteristics of numerical computations. I only listed the names of the characteristics.

So let’s learn in brief about those characteristics for getting a better understanding of this chapter. If you haven’t read the previous article then make sure to read it before reading this one.

**Characteristics of Numerical Computing**

- Accuracy
- Rate of convergence
- Numerical stability, and
- Efficiency

#### Accuracy

Accuracy refers to how well a method matches the expected result. It can also be interpreted as the degree to which a given quantity is correct and free from error. For example, quantity specified as 100±1 has an (absolute) accuracy of ±1 (meaning its true value can fall in the range 99-101), while a quantity specified as 100-+2 has an accuracy of ±2 meaning its true value can fall in the range 98-102.

#### Rate of convergence

Rate of convergence refers to how long it takes to apply a given method for a given time- step. Suppose that the sequence {xₖ} of values converges to a number L.

#### Numerical stability

Numerical stability refers to how well the method copes with stiff constraints before error becomes unacceptable large. Numerical stability is considered to be a desirable property of numerical algorithms.

Its precise definition depends on the context and is related to the accuracy of the algorithms. Depending on the specific computational method small errors, instead of damped, can be magnified leading to large errors

#### Efficiency

Efficiency refers to the number of steps in the algorithm, the computer time, and the amount of the computer (or computing device) that is required.

### Exercise – 19.2

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