95503 統資軟體課程講義銘傳大學應用統計資訊學系蔡桂宏製作 1...

銘傳大學應用統計資訊學系蔡桂宏製作

1

9550395503 統資軟體課程講義統資軟體課程講義

符號數學運算軟體符號數學運算軟體


2

9550395503 統資軟體課程講義統資軟體課程講義

MAPLEMAPLE 軟體簡介軟體簡介


3

Introduction to Maple

Installing Maple Starting Maple The Maple Worksheet Window Accessing Help Pages Entering Expressions in Maple


4

Maple worksheet window


5

Maple worksheet window

A.toolbar

B.Context barC.Section heading

D.Maple inputE.Maple output

F.Execution group

G.Section range bracket

H.Prompt

I.Symbol palette

J.Expression palette

K.Matrix palette

L.Vector palette


6

Maple Help window


7

Maple Help window

A.Help page nameB.toolbar

C.Help navigator Tabs

D.Topic folder

E.Bullet

F.Description

G.Help Page Title

H.Example

I.Also See


8

Maple 9 Getting Started Guide

This guide contains an introduction to the graphical user interface and a tutorial that outlines using Maple to solve mathematical problems and create technical documents. It also contains additional information for new users about the help system, New User’s Tour, example worksheets and Maplesoft Web site.


9

Maple 9 Learning Guide

This guide explains how Maple and the Maple language work. It describes the most important commands and uses them to solve technical problems. User hints for Maplet applications are also described in the guide.


10

Maple 9 Introductory Programming Guide

This guide introduces the basic Maple programming concepts, such as expressions, data structures, looping and decision mechanisms, procedures, input and output, debugging, and the Maplet User Interface Customization System.


11

Maple 9 Advanced Programming Guide

This guide extends the basic Maple programming concepts to more advanced topics, such as modules, graphics programming, and compiled code.


12

Tutorials

Maple provides three tutorials that can be accessed from the New User submenu of the help menu: Full Tour Quick Tour Basic How To


13

Websites

www.maplesoft.com Information on products, support, and services.

www.mapleapps.com Including a forum for sharing solutions, demon

strations, of Maple PowerTools, and an online tutorial

www.maple4students.com Includes course help, Maple tutorials, and Ma

ple graphics


14

Tokens—Reserved Words(1)

Keywords Purpose

break, next loop control

if, then, elif, else if statement

for, from, in, by, to,for and while loops

while, do 　proc, local, global, option, error,

procedures

return options, description

　

export, module, use modules

end ends structures


15

Tokens—Reserved Words(2)

Keywords Purpose

assuming assume facility

try, catch, finallyexception handling

read, saveread and save statements

quit, done, stop ending Maple

union, minus, intersect, subset

set operators

and, or, not, xor Boolean operators

impliesimplication operator

modmodulus operator


16

Tokens—%operator

% last expression %% second-last expression %%% third-last expression


17

Tokens—Binary Operator

Operator

MeaningOperator

Meaning

+ addition < less than

- subtraction <= less or equal

* multiplication > greater than

/ division >= greater or equal

^ exponentiation <> not equal

$ sequence operator -> arrow operator

@ composition union set union

@@ repeated composition minus set difference

&string neutral operatorintersect

set intersection

, expressiop separator ::type declaration, pattern binding

|| concatenation and logical and

.noncommutative multiplication or logical or

.. ellipsis xor exclusive or

mod modulo implies inplication:= assignment subset subset


18

Tokens—Unary Operator

Operator

Meaning

+ unary plus(prefix)- unary minus(prefix)! factorial(postfix)$ sequence operator(prefix)not logical not(prefix)&string neutral operator(prefix). decimal point(prefix or postfix)%integer

label(prefix)


19

Tokens—Initially known Name

Name Meaning Name Meaning

Catalan

Catalan's constantlasterror most recent error

constants

initially-known symbolic const.

libname

pathname of Maple library(ies)

Digitsnumber of digits in floating-point

NULLempty expression sequence

FAILcannot determine value

Ordertruncation order for series

false Boolean evaluation Pi mathematical constant

gamma

Euler's constantprintlevel

control display of information

I complex number true Boolean evaluation

infinity

mathematical infinityundefined

undefined


21

Token – Escape Characters

? – Help procedure ! --Host operating system # -- Comment \ --continuation of lines and grouping

characters in a token.


22

Selection Statements

if Boolean expression then statement sequenceelif Boolean expression then statement sequenceelse statement sequenceend if


23

Repetition Statements

for name from expr by expr to expr while expr do

statement sequence

end do;


24

1.1 Introduction

Understanding functions in Maple is also a good starting point for our discussions about Maple programming.

There are two distinct ways to represent mathematical functions in Maple. That is Maple expression ( := ) and a Maple function ( -> ).


25

1.2 Functions in Mathematics

the formula has as its domain the set of all real numbers, its codomain is the set of all positive real numbers.

the formula has as its domain the set of all positive real numbers, its codomain is the set of all positive real numbers.

the formula has as its domain the set of all negative real numbers, its codomain is the set of all positive real numbers.

2)( xxf

2)( xxg

2)( xxh


26

the function f is not invertible.

The inverse of the function g is .

The inverse of the function h is .

So f, g, and h all have the same formula (i.e., rule) but they are not the same function. The domain and codomain are important parts of the definition of a mathematical function.

xxg )()1(

xxh )()1(


27

1.3 Functions in Maple

These two ways of representing mathematical functions are not equivalent. And it is subtle and non-obvious.

A Maple function is something defined using arrow notation ( ->).

A Maple expression is something defined using ( := ).x -> x^2;x -> a*x^2;(x,a) -> a*x^2;


28

. Maple will treat all unassigned names (i.

e., all unknowns) as variables.

The mathematical function . g := x -> x^2 – 1; g(2);

g(x) := x^2-1; g(2);

cbxax 2

1)( 2 xxg


29

g := cos + ln; k := x -> cos(x) + ln(x); g(Pi); k(Pi);

h := cos + (x -> 3*x-1); h(z);


30

f := x -> (1 + x^2)/x^3; g := (1 + (x -> x^2))/(x -> x^3); h := (1 + (z -> z^2))/(y -> y^3);

f(1); g(1); h(1);

m := x -> (1 + exp(x))/x^3; n := (1 + exp)/(x -> x^3);

m(1); n(1);


31

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

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


32

1.4 Expressions vs. functions: Some puzzles

The following examples are meant to show that there are still a lot of subtle things to learn about variables and functions and how Maple handles them.


33

Puzzle 1

f1 := x^2+1; f2 := y^2+1; f3 := f1 + f2; f3 is a function of two variables.

g1 := x -> x^2+1; g2 := y -> y^2+1; g3 := g1 + g2; g3(x); g3 is not a function of two variables.


34

Puzzle 2

x:='x': a:=1: b:=2: c:=3: a*x^2+b*x+c; f := unapply( a*x^2+b*x+c, x ); g := x -> a*x^2+b*x+c; f(x); g(x); D(f); D(g);

p := x^2 + sin(x) + 1; p(2); p := unapply(p,x); p(2);


35

Puzzle 3

plot( x^2, x=-10..10 );

plot( x->x^2, -10..10 );

plot( x^2, -10..10 ); plot( x->x^2, x=-10..10 );


36

x := 5;

plot( x^2, x=-10..10 );

plot( x->x^2, -10..10 );


37

Puzzle 4

f := x^2; f := x*f; f;

g := x -> x^2; g := x -> x*g(x); g(x);


38

Puzzle 5

x^2; f := %; plot( f, x=-3..3 );

x^2; g := x -> %; plot( g, -3..3 );


39

1.5 Working with expressions and Maple functions (review)

g := x -> x^2-3*x-10; g(x); g; print(g); eval(g); op(g);


40

plot( a*x^2, -5..5 );

plot( x->a*x^2, -5..5 );

plot3d( (x,a)->a*x^2, -5..5, -10..10 );


41

We wanted to evaluate our mathematical

function at a point, say at 1. f := x^2 - 3*x-10; g := x -> x^2-3*x-10;

subs( x=1, f ); eval( f, x=1 ); g(1);

f(1); subs( x=1, g );


42

eval( f, x=1 ); subs( x=1, f ); Think of reading eval( f, x=1) as " evalua

te f at x=1" and think of reading subs (x=1,f) as " substitute x=1 into f".

factor( f ); factor( g(x) );

factor( f (x) ); factor( g );


43

diff( f, x ); D( g ); diff command needed a reference to x in it

but the D command did not.

D( f ); diff( g, x );


44

Let us do an example of combining two mathematical functions f and g by composing them to make a new function .

For the expression:

f := x^2 + 3*x; g := x + 1; h := subs( x=g, f ); subs(x=1, h);

))(()( xgfxh


45

For the function:

f := x -> x^2 + 3*x; g := x -> x + 1; h := f@g; h(x); h(1);


46

Example: representing a mathematical function of two variables.

f := (x^2+y^2)/(x+x*y); g := (x,y) -> (x^2+y^2)/(x+x*y); subs( x=1, y=2, f ); eval( f, {x=1, y=2} ); g(1,2);

simplify( f ); simplify( g(x,y) );


47

Here is how we compute partial derivatives of the expression.

diff( f, x ); simplify( % ); diff( f, y ); simplify( % );

D[1](g); simplify( %(x,y) ); D[2](g); simplify( %(x,y) );


48

1.6 Anonymous functions and expressions (review) Here we define an anonymous function an

d then evaluate, differentiate, and integrate it.

(z -> z^2*tan(z))(Pi/4);

x -> x^3 + 2*x;

%(2);

D( %% );

int( (%%%)(x), x );


49

These next two commands show again that defining a function and naming a function are two distinct steps.

z -> z/sqrt(1-z);

f := %;


50

f := ((x,y) -> x^2) + ((x,y) -> y^3); f(u,v); f(2,3);

g := (x -> x^2) + (y -> y^3); g(u,v); g(2,3);


51

plot(w^3+1, w=-1..1);

plot( ((x,y)->x^3-y^3)(w,-1), w = -1..1 );

plot( w->(((x,y)->x^3-y^3)(w,-1)), -1..1 );


52

1.7. Functions that return a function

f := a -> ( y->a*y ); f(3); f(3)(4);


53

f := (x,y) -> 3*x^2+5*y^2; f(x,3);

fx3 := x -> f(x,3); fx3(x);

slice_f_with_y_fixed := c -> ( x->f(x,c) ); fx3 := slice_f_with_y_fixed(3); fx3(x);

95503 統資軟體課程講義 銘傳大學應用統計資訊學系 蔡桂宏 製作 1...

Documents

95503 統資軟體課程講義銘傳大學應用統計資訊學系蔡桂宏製作 1...