MAT 4725Numerical Analysis

Section 1.4

Loops with “do” statements

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Homework

Download homework from the web Read

• 2.1.4 while-do loop

• 1.6.1 documentations

• 1.6.2 format printing

Quiz on 1.6.2, we will not lecture on that section

Preview

Monotonic Sequence Theorem (Stewart, section 12.1)

Introduce the first type of repetition statements – the for loop

Allow a specific section of code to be executed a number of times

Introduces simple arrays

Definition

A sequence {an} is bounded above if M such that

anM n

A sequence {an} is bounded below if m such that

anm n

Monotonic Sequence Theorem

The following sequences are convergent Increasing and bounded above Decreasing and bounded below

1n na a

Example

Show that the sequence defined by

is convergent and find its limit.

1 1

12 and for 1

3nn

a a na

Example

From homework 01, we know

1 1

12 and for 1

3nn

a a na

10 2 and for n n na a a n

Zeng Section 1.4

Please listen to the explanations before you type in the program.

It takes one minute to explain.

Example 1 Print the square of the first 10 positive

integers What is the task being repeated?

Example 1

Example 1

i

1 2 101 4 100

i2i

Example 1

> sq();149

Structure of the for loop

Structure of the for loop

Example 2 Print the square of the first 10 positive

odd integers

Example 2

Example 2

> sq2();19

25

Example 3 Print the square of the first n positive

integers

Example 3 Print the square of the first n positive

integers Introduces array and seq Note that these commands are not

necessary here

Example 3

Example 3

[ ]x n

[3]x[2]x[1]x

Example 3

> sq3(2);1, 4

> sq3(5);1, 4, 9, 16, 25

Example 4

Fibonacci sequence is defined by

0 1 1 20, 1, for 2,3,

{0, 1, 1, 2, 3, 5, }

k k kF F F F F k

Example 4 Write a program that generate the first

n+1 terms of the Fibonacci sequence

F0,F1,…,Fn

0 1 1 20, 1, k k kF F F F F

Example 4 0 1 1 20, 1, k k kF F F F F

Example 4 0 1 1 20, 1, k k kF F F F F

What happen if we do not

initialize F?

Example 4 0 1 1 20, 1, k k kF F F F F

Why there is no print statement?

Example 4 0 1 1 20, 1, k k kF F F F F

Example 5

2 1 2 1

0 0

( 1) ( 1)sin

(2 1)! (2 1)!

k knk k

k k

x x xk k

Write a program, for the input of x and n, to approximate the value of sin(x) by the first sum of the first n+1 terms in the Taylor series.

Example 5

2 1

0

( 1)sin

(2 1)!

knk

k

x xk

Example 5

2 1

0

( 1)sin

(2 1)!

knk

k

x xk

Example 5

2 1

0

( 1)sin

(2 1)!

knk

k

x xk