MAT 4725 Numerical Analysis Section 1.4 Loops with “do” statements

Homework Download homework from the web Read while-do loop documentations 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 {a n } is bounded above if  M such that a n  M  n A sequence {a n } is bounded below if  m such that a n  m  n

Monotonic Sequence Theorem The following sequences are convergent Increasing and bounded above Decreasing and bounded below

Example Show that the sequence defined by is convergent and find its limit.

Example From homework 01, we know

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 > sq(); 1 4 9

Structure of the for loop

Example 2 Print the square of the first 10 positive odd integers

Example 2

> sq2();

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

> sq3(2); 1, 4 > sq3(5); 1, 4, 9, 16, 25

Example 4 Fibonacci sequence is defined by

Example 4 Write a program that generate the first n+1 terms of the Fibonacci sequence F 0,F 1,…,F n

Example 4

What happen if we do not initialize F?

Example 4 Why there is no print statement?

Example 4

Example 5 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