1. CS 101: Numerical Computing Abhiram Ranade

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 = + 2 31 -1 Min negative number: H H...H = - 2 31 -1 2 31 is almost = 2 * 10 9 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 2 63 -1, about 10 19 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 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); 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 Make pattern go around a polygon/circle Number of times to spiral: read from cin

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); }

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 1 x

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 x n given x and n. Write a program that computes n! given n.