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Goal

The goal of this assignment is to introduce Floating Point operations in MIPS.

Background

Newton devised a method to aproximate a root of an equation by repeatedly apply the formula until the error was sufficiently small. This formula, also known as the Newton–Raphson method, is defined as:

𝑥𝑛+1 = 𝑥𝑛 − (𝐹(𝑥𝑛)/𝐹′(𝑥𝑛))

Consider a continuous function F(x)=x3-4x2+1 with its derivative F’(x)=3x2-8x. Thus, by calculating successive values of x, the value of F(xn) will approach 0, and hence be a root of F.

The Assignment

Write a MIPS program which will find a value of xn such that F(xn)=0, to a tolerance of 10-6.

Prompt the user to enter an initial integer value for x, hence x1. Then calculate each new x value, using the formula above, until F(x) is sufficiently close to 0. Try the value of 2 as a test input value.

All calculations must be done as doubles. Your program should output x and the value of F(x) at each step and finally print the root (x) that you found. You should try other values for the initial value of x to test your program.

Note, your output will not be so clean since the default print of a double includes many decimal places and could be in scientific notation, don’t worry about aligning the output.

The output with an input of 2 should look like the following:

Please enter an integer: 2 x=2.0, F(x)=-7.0

x=0.25, F(x)=0.765625

x=0.6724137931034483, F(x)=-0.5045358563286728 x=0.5469975054591968, F(x)=-0.033160000071302154 x=0.5374642805088174, F(x)=-2.1525861187732254E-4 x=0.5374015797596746, F(x)=-9.386846855363729E-9 x=0.5374015797596746

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