CS 101: Numerical Computing

1. Abhiram Ranade CS 101: Numerical Computing

2. Representing Integers • “int x;” : reserves one cell in memory for x. • One cell: “One word” has 32 capacitors. Each capacitor can have High/Low charge. • Storing positive numbers only: Represent number to be stored in binary, and store High charge for 1, Low charge for 0. • How to store negative numbers?

3. 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 • 231 is almost = 2 * 109 • Actual representation is slightly different.

4. How to store larger numbers? • Use Long integers: “ long x;” Reserves 2 words, with one sign bit. Max magnitude 263-1, about 1019 • Use floating point representation.

5. 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.

6. Summary • Cohoon gives exact details. Not important in 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 the work of man.” -- Leopold Kronecker.

7. 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 in the 23rd digit. • “float” stores only 7 digits. 23rd digit ignored. -- “roundoff error”. • w = avogadro!

8. 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!

9. Simple Numerical Program #include <turtlesim.h> procedure turtleMain(){ float c; cout << “Centigrade temperature:” << endl; cin >> c; cout << “Fahrenheit:” << (c*9/5)+32; }

10. 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.

11. 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.

12. 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..

13. Reassignment in Loops int i=1; repeat(5){ forward(i); right(90); forward(i); right(90); i = i + 1; } • What will be drawn?

15. Nested Squares How will you draw this pattern? Can you avoid writing 7 calls to procedure Polygon? Side length: 2, 4, 6, ...

16. Homework: draw this • Number of times to spiral: read from cin • Make pattern go around a polygon/circle

17. 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

18. Spiral using for int i; for(i=1; i < 6; i=i+1){ forward(i); right(90); forward(i); right(90); }

19. for(xxx; yyy; zzz) { www } 0. Execute “xxx;”. Must be an assignment statement. “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.

20. Condition • “ a op b” where op is <, >, <=, >=, ==, != • == : is equal to • != : is not equal to • Conditions can be combined: • (a > 0) && (a <= 9) : true if a is a positive digit. • && : conjunction. “and” • || : disjunction. “or”

21. More examples of for • for(j=10; j>0 ; j=j-1){ cout << j*j << endl; } What is printed? • for(int i=1; i < 100; i=i*2){ cout << i << endl; } What is printed? i can be defined inside for – then i cannot be accessed outside of loop body ( {cout... } )‏ { ... } is the scope of i.

22. And one more.. int i=0,num; for( cin >> num; num > 1; num = num/2){ } cout << i << endl; num=num/2 : integer part of the quotient. What is printed?

23. Computing ln x • Must use arithmetic operations. • Estimate the area under f(x) = 1/x from 1 to x. • Area approximated by small rectangles.

24. Riemann Integral x 1

25. 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)‏

26. Program to compute ln procedure turtleMain(){ float x, area=0, w; cin >> x; w = (x-1)/1000.0; for(int i=1 ; i <= 1000 ; i=i+1){ area = area + w*(1/(1+(i-1)*w); } cout << “ln(” << x << “)=” << area << endl; }

27. Notes • ith iteration adds area of ith rectangle to “area”. • It is customary to count starting at 0, so there 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

28. New program procedure turtleMain(){ 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 << “ln(”<< x << “)=”<< area << endl; }

29. Homework • Write a program that prints a conversion table from centigrade to Fahrenheit; for i = 1 to 100 it should print equivalent value in Fahrenheit. Use above procedure. • Write a program that integrates f(x)=x. Check how much error it makes by doing a direct calculation. • Write a program that computes xn given x and n. • Write a program that computes n! given n.