CS107: Computing for Math CS107: Computing for Math and Science - - PowerPoint PPT Presentation

CS107, Prof. Steinberg, f10

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Lecture 23

CS107: Computing for Math CS107: Computing for Math and Science and Science

Lecture 24: Maple

CS107, Prof. Steinberg, f10

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Lecture 23

Maple Maple

• Running maple via internet:

Like matlab but

– Use alfalfa.rutgers.edu in place of remus .rutgers.edu – Use xmaple in place of matlab – To exit: quit

CS107, Prof. Steinberg, f10

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Lecture 23

Running Maple Running Maple

• Or run Maple in a computer lab
• Or buy it and run on your own computer:
• Students may purchase their own Maple

Student Edition for $59 by going to: http://webstore.maplesoft.com

– Requires proof of student status. – Choose a Category: Student – Select your Location: US – Promotional Code: RUTGERS72 – Click: Start Shopping

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Lecture 23

? To access help system ? To access help system

• EG ?+ or ?factor
• Or click on ? Icon at top right of window
• Calling Sequence, Params, Description,

Examples

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Lecture 23

Numbers Numbers

• Integers - of any size
• Real - of any precision

– E.g. Digits := 20;evalf(Pi);

20 3.1415926535897932385

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Lecture 23

Variables Variables

• Use := for assignment

x:=1+2;

• But variables don’t have to have values

x:= a + b;

• So Maple combines variables

– as in program: store a value – As in math: represent an unspecified value

• = means an equation, not assignment

x = a + b does not change value of x – Can even do x := c = a + b

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Lecture 23

Variables Variables

• To “clear” a variable means to remove

any value it might have

• Clear all variables with

restart;

• Clear, e.g., x with

unassign(‘x’); Note ‘ ‘

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Lecture 23

Variables Variables

> a:=b+2; a := b + 2 > a; b + 2 > b:=3; b := 3 > a; 5 > b:=4; b := 4 > a; 6 > unassign('b' ); > a; b + 2

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Lecture 23

But Compare But Compare

• a := b + 2;
• b := 1;
• a

3

• b := 2;
• a

4

• y := 1;
• x := y+2;
• x

3

• y :=2;
• x

3

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Lecture 23

But Compare But Compare

value in a b a := b + 2; b+2 b := 1; 1 a 3 b := 2; 2 a 4 value in x y y := 1; 1 x := y+2; 3 x 3 y :=2; 2 x 3

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Lecture 23

Evaluating variables Evaluating variables

• Top-level variables (not local to a proc)

are evaluated all the way until no variable has a value

– Eg i := h; h := f+g; f := 3; sets i to 3+g

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Lecture 23

Dittos Dittos

• %
• %%
• %%%

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Lecture 23

Symbolic expressions Symbolic expressions

• Value of a variable can be an expression

a := y= x^2-2*x+1;

• Maple has functions that operate on expressions:

factor(a); returns y = (x - 1)2

• The function op returns pieces of an expression
• p(1, a) returns y
• p(0, a) returns =
• p(a) returns y, x^2-2x+1
• A sequence is written with commas a, b, c
• If s is a sequence, s[j] is j’th element of s
• p(a)[2] is x2-2x+1

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Lecture 23

Ranges Ranges

• 1 .. 3 means 1, 2, 3

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Lecture 23

Sequences Sequences

• seq(expression, var=range)

seq(j^2, j=2..5) is 4, 9, 16, 25

• p(expr)
• p(a*x^2+b*x+c) is a*x^2, b*x, c
• nops(expr)
• 2:5

1, 2, 3, 4, 5

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Lecture 23

Lists & Sets Lists & Sets

• Lists: [ seq ]
• Sets: { seq } like lists but no repetition or order

> evalb([2, 3, 3] = [2, 3]); false > evalb({2, 3, 3} = {2, 3}); true > evalb([3, 2] = [2, 3]); false > evalb({2, 3} = {2, 3}); true

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Lecture 23

Solving Equations Solving Equations

• solve(x=2*a^2/b, b);
• solve(a*x^2+b*x+c=0, x)
• solve({a*x+b*y=c,d*x+e*y=f},{x,y});
• Assign

s = solve(...); assign(s); Assigns solution values to variables

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Lecture 23

Functions Functions

• Expression vs function

x^2+y^2 is an expression

• an expression is a computation or an

equation (x,y)->x^2+y^2 is a function

• a function is a mapping from 1 or

more arguments to an expression

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Lecture 23

apply apply

• apply(function, arg)
• Result is an expression

apply(sqrt, 4) 2 apply(foo, a, b, c) foo(a, b, c) f:= (x)-> b+2*x^3 apply(f, a) b+2*a^3

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Lecture 23

unapply unapply

• unapply(expression, variables)

Result is a function from variables to expression f:= unapply(b + 2*x^3, x) x->b+2*x^3 f(d) b+2*d^3

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Lecture 23

Functions as data Functions as data

• Stored as value of a variable

f := x-> x^2+2; f(3);

• Used as arguments and values of functions

unapply(x^2+2, x); D(f);

map(f, [2, 4, 5])

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Lecture 23

Differentiating Differentiating

• D(f) is f’, I.e. a function

– E.g. D(cos) is sin

• diff(a,x) is derivative of expression a with

respect to x

– E.g., diff(cos(x), x) is sin(x)

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Lecture 23

Finding maxima Finding maxima

• f := x -> sin(x) * cos(x)
• D(f)

x->cos(x)^2-sin(x)^2

• solve(D(f)(x)=0,x)

1/4 pi, 3/4 pi

• See cannon.mw

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Lecture 23

Programming Programming

• If

if x > y then z:= z elif x = y then z := 0 else z := y end if

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Lecture 23

Programming Programming

• If

if x > y then z:= z elif x = y then z := 0 else z := y

end if

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Lecture 23

Programming Programming

• For

for j from 1 by 2 to 9 do x:=x+j; end do;

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Lecture 23

Programming Programming

• proc(vars)

end proc

• read “filename.mpl” to read in definitions

from a file.