cs: numerical computing

8/11/2019 CS: Numerical Computing

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CS 101: Numerical Computing

Abhiram Ranade


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Sign magnitude representation

Key idea: One of the capacitors determines sign.

L = +, H= -

x = 5; : store L LLLL...LHLH x = -5; : store H LLLL...LHLH

Max positive number: L H...H = + 231-1

Min negative number: H H...H = - 231

-1 231is almost = 2 * 109

Actual representation is slightly different.


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Floating Point Representation

float y; : reserve 1 word = 32 bits as before.

24 bits : mantissa. 8 bits: exponent. Each

includes a sign bit. Mantissa can only store 23 bit accuracy, about 7

decimal digits.

Exponent can be between -100 and +100 (about).

double z; : 53 bits of mantissa, 11 bits exponent.


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Summary

Cohoon gives exact details. Not importantin this course. Various other formats, e.g.

unsigned int, short. Circuits are more complex for floating

representation, sign...

God made natural numbers, the rest is thework of man. -- Leopold Kronecker.


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Floating Point Arithmetic

float w, y=1.5, avogadro=6.023 E 23 ;

w = avogadro + y;

What is the value of w?

Exact value of w and avogadro will differ inthe 23rd digit.

float stores only 7 digits. 23rd digitignored. -- roundoff error.

w = avogadro!


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Mixed Arithmetic

C++ converts from float to int and vice

versa as needed.

int x=10; float y=6.5;

x=y; : 6.5 is rounded down. x=6.

360/x : integer result is calculated.

360.0/x : float result is calculated.

360/y : float result. Try out yourself!


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Simple Numerical Program

#include

procedure turtleMain(){

float c;

cout c;cout


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Remarks on Arithmetic

Multiplication: write explicitly using *.

*, / have higher precedence than +, -

*, / have equal precedence, evaluated in left

to right order. Similarly +, -

x = m % n; % : mod. m=10,n=4, x=2.

Same precedence as *, /

Use () to override default precedence.


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Comments

Programs are read by compilers, but also by

other programmers.

You should include extra description

(comment) to help other programmers

figure out the logic:

Style 1: // ...... to end of line

Style 2: /* .... */ Examples soon.


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Important Idea: Reassignment

int m=5;

m = 3*m + 1; // Reassignment

Mathematically absurd: simplify to

0 = 2*m+1

C++ interpretation: Calculate the value of rhs,then store it in lhs. Above: m = 16.

Important inside loops. Next..


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Reassignment in Loops

int i=1;

repeat(5){

forward(i); right(90);


i = i + 1;}

What will be drawn?


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Nested Squares

How will you draw this pattern? Can you avoid

writing 7 calls to procedure Polygon?

Side length: 2, 4, 6, ...


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Homework: draw this

Make pattern go around a polygon/circle

Number of times to spiral: read from cin


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Common Programming pattern

Repeat n times with i taking different values

Cleanly specified using for statement:

for( xxx; yyy; zzz ) { www } // next..

Useful in numerical computing also


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Spiral using for

int i;

for(i=1; i < 6; i=i+1){



}


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for(xxx; yyy; zzz) { www }

0. Execute xxx;. Must be an assignmentstatement. loop initialization

1. Evaluate condition yyy. loop test. If true,

continue with step 2; if false, for statement

execution ends.

2. Execute www. loop body

3. Execute zzz. loop increment

4. Continue from 1.


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Condition

a op b where op is , =, ==, !=

== : is equal to

!= : is not equal to

Conditions can be combined:

(a > 0) && (a


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More examples of for

for(j=10; j>0 ; j=j-1){ cout


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And one more..

int i=0,num;

for( cin >> num; num > 1; num = num/2){ }

cout


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Computing ln x

Must use arithmetic operations.

Estimate the area under f(x) = 1/x from 1 to

x.

Area approximated by small rectangles.


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Riemann Integral

1 x


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How many rectangles?

More the merrier! Say 1000.

Total width of rectangles = x - 1.

Width w of each = (x - 1)/1000

x coordinate of left side of ith rectangle

1 + (i-1)w. Height of ith rectangle = 1/(1+(i-1)w)


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Program to compute ln


float x, area=0, w;

cin >> x; w = (x-1)/1000.0;

for(int i=1 ; i


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Notes

ith iteration adds area of ith rectangle to

area.

It is customary to count starting at 0, sothere is a zeroth rectangle, first rectangle..

height of (new) ith rectangle = 1+ iw

i++ : short form for i=i+1

area += q : short form for area = area + q


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New program


float x, area=0, w;

cin >> x; w = (x-1)/1000.0;

for(int i=0; i < 1000; i++)

area += w/(1+i*w);

cout


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Homework

Write a program that prints a conversion tablefrom centigrade to Fahrenheit; for i = 1 to 100 itshould print equivalent value in Fahrenheit. Use

above procedure.

Write a program that integrates f(x)=x. Checkhow much error it makes by doing a directcalculation.

Write a program that computes xngiven x and n.

Write a program that computes n! given n.

cs: numerical computing

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