youngjoo chung - tutorial on symbolic computing with mathematica [2011] [p47]

8/10/2019 Youngjoo Chung - Tutorial on Symbolic Computing With Mathematica [2011] [p47]

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Tutorial

on

Symbolic Computing

withMathematica

Youngjoo Chung

School of Info. and Comm., GIST

[email protected]

2011. 7. 7


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The concept of Mathematicais to create once and for all a single system that could handle

all the various aspects of technical computing in a coherent and unified way.

Manipulation of the very wide range of objects involved in technical computing using only

a fairly small number of basic primitives (over 3,000 built in the kernel of version 7). The functionality can be easily extended through user-defined functions and external

programs.

Platform-independent interactive documents known as notebooks (.nb files)

System-provided macros and user-defined macros in packages (.m files)

Interactive and non-interactive sessions

Communication with external programs using MathLink

Mathematicacomputing environment

TCP/IP

mle.exeMathLink

Standard Packages

Mathematica

Kernel

Windows/Unix

Platform

Windows/Unix

Platform

Add-On Packages

User Packages

Other Packages

User Frontend

Numerical

Analysis

Control

Instrumenta

External applications are written in C/C++.

Mathematicais used for both data input and output.

External applications andMathematicacommunicate via MathLink.

The functionality can be extended through user-defined functions and external programs.

More information available at http://www.wolfram.com

Mathematica Tutorial.nb 3


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Algebraic Calculation

Algebraic Calculations (MathematicaTutorial)

Symbolic Calculations (MathematicaTutorial)

Expand

Factor

Apart

Together

Simplify

Basic Algebra

Polynomial Algebra (MathematicaGuide)

Polynomial Systems (MathematicaGuide)

Elementary Functions

Elementary Functions (MathematicaGuide)

Trigonometric Functions

Trigonometric Functions (MathematicaGuide)

Trigonometric Expressions (MathematicaTutorial)

Complex Variables

Complex Numbers (MathematicaGuide)

Functions of Complex Variables (MathematicaGuide)

Flow Control

Conditionals (MathematicaGuide)

Loops and Control Structures (MathematicaTutorial)

Flow Control (MathematicaGuide)

Formula Manipulation

4 Mathematica Tutorial.nb


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Formula Manipulation

Formula Manipulation (MathematicaGuide)

Manipulating Equations (MathematicaGuide)

User Package Functions

Mathematica provides a solid foundation for symbolic computing and a number of support-

ing functions.

Built-in kernel functions

User-defined functions to complement the built-in kernel functions

A mechanism is needed to facilitate the algebraic manipulation of mathematical expressionswith the following properties:

Seamless integration with the computing environment of Mathematica

Selective targeting of the object to apply functions

Improved handling of subscripts (a1), tildes (), hats (), etc.

On-line setting and clearing of attributes

Addition of comments

Improved handling of derivatives, integrals and summations

Minimal use of variables

Algebraic manipulation of formulas using symbolic computing

Mathematica is not a word processor or an equation editor.

Application of functions

Substitution using mathematical identities

Allows focusing on the principles instead of time-consuming and error-prone calculations.

Good readability

Minimization of human errors during calculations

Expressions that closely resemble the traditional mathematical style, e.g., subscripts and

vector notations

A large collection of user-defined functions

Basic Algebra

Differentiation

Integration

Summation


Complex Variables

Vectors and Matrices Polynomials and Series

Functional Analysis

Equations

Operator Analysis

Plotting



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A large collection of user-defined functions

Basic Algebra

Differentiation

Integration

Summation


Complex Variables

Vectors and Matrices

Polynomials and Series

Functional Analysis

Equations

Operator Analysis

Plotting

Etc.

MPMAF

An abbreviation ofMPMapApplyFunc

The platform for algebraic manipulation of all or parts of an expression

User-defined functions

The function names start with the prefix MP.

Algebraic manipulation of expressions

Mathematical identities Can be used separately independent of the macro MPMAF, in which case the features

provided by the options of MPMAFcannot be used.

Basic Algebra

Expansion

ExpandAHx+ aL2E

a2 +2 a x+ x2

The power exponent must be an integer. Otherwise, useMPExpandBinomial.

MPExpandBinomial@Hx+aLn, kDMPEvalSum@%D

k=0

n ak x-k+n n !

k !H- k +nL!Ha+ xLn



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MPExpandBinomial@Ha+ b +cLn, k, lDMPEvalSum@%DPowerExpand@%D

k=0n

l=0-k+n ak bl c-k-l+n n !

k !l !H- k -l + nL!

cnb +c

c

na+ b +c

b+ c

n

Ha+ b +cLn

Factoring

FactorAx2 +Ha+ bLx +a bEHa+ xL Hb +xL

MPFactor

MPFactor@a+ b, aD

a 1+b

a

Reduction of fractions

x2

x+ a

Hx +aL2 -a2 - 2 aHx+ aL +2 a2

x+ a

x2

a+ x

- a2 -2 a x +Ha+ xL2a+ x

ApartB x2

x+ a, xF

- a +x +a2

a+ x

MPApart

MPApartB sx2 ex

n2 e0 - ex+

sy2 ey

n2 e0 - ey+

sz2 ez

n2 e0 - ez,9ex, ey, ez=F

%. - sx2 - sy2 -sz2 - 1

- sx2 - sy

2 -sz2 -

n2 sx2 e0

- n2 e0+ ex

-

n2 sy2 e0

- n2 e0+ ey

-n2 sz

2 e0

- n2 e0+ ez

- 1 -n2 sx

2 e0

- n2 e0+ ex

-

n2 sy2 e0

- n2 e0+ ey

-n2 sz

2 e0

- n2 e0+ ez

Separation of variables for solving differential equations



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y

x 2 x y

MPMAFB%, MPSepVars,8All, x, Side Right,

MPEvalInt, All,

MPSolve,8All, y


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MPTrigToDoubleACos@xD2, CosE1

2H1+ Cos@2 xDL

Complex Variables

Asterisk (*) is interpreted as the complex conjugate

MPComplexExpandAHu+vL*, u, TargetFunctions ConjugateE

v+u*

The built-in function ComplexExpand gives a rather different (even though equivalent)

result.

ComplexExpand@Conjugate@u+vD, u, TargetFunctions ConjugateD

v+

1

2Hu+Conjugate@uDL

CoshB 12

Hu -Conjugate@uDLF- v+ 12 Hu+Conjugate@uDL SinhB 12

Hu- Conjugate@uDLF

The exponential form of a complex number

MPComplexToExp@x+ yD

ArcTanA y

xE

x2 +y2

Polynomials and Series

Eliminate the second highest order term

MPMergePolyAx3 +a x2 +c, xE

-a3

27+c -

a2 x

3+

a

3+x

3

Transform a polynomial by making a replacement of the variable

MPTransPolyAx3 +a x2 +c, x, a +xE

c+ a2 Ha+ xL- 2 aHa +xL2 +Ha +xL3

A case of two variables

MPTransPolyAx2 Hy +aL,8x, y


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MPTaylorB 1 +x , x, 3F

1+x

2-

x2

8+

x3

16

MPTaylorB 1 +Hx+ DxL2 , Dx, 1F

1 +x2 +xDx

1+ x2

MPAFE@y@xD, MPTaylor,8All, x- x0, 2, Variables x


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Differentiation

The built-in derivative operator D, or , evaluates the expression immediately.

D@f@xD, xDf@xD

The total derivative operator Dtassumes all symbols are dependent variables.

Dt@a f@xD, xDDt@a, xDf@xD+ a f@xD

The partial differential operator

xdelays the evaluation until the commandMPEvalDor

MPExpandDis encountered.

y 3

x2 yHa f +b gL

MPMAF@%, MPExpandD,8At@2D,8f, g


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Transformation of variables

MPTransDB 2 f

w2, VT :w, l, w 2pc

l>F

l3

2 c2 p2

f

l+

l4

4 c2 p2

2 f

l2

Evaluation of vector operators

MPAFEA2 y, MPEvalVecOps,8All, y y@r, jD, Cylindrical@r, j, zDF

2 y 1

r

y

r+

2 y

r2+

1

r2

2 y

j2

2 y 1

r2

2 y

j2+

1

r

rr

y

r

MPAFEA2 y, MPEvalVecOps,8All, y y@x, y, zD, Cartesian@x, y, zD


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F HF f x

F f x

Combining multiple integrals

KHF f@xD xO KHF g@xD xOMPCombInt@%, RV 8x, x

:f@xD F@kD k x k, g@xD G@kD k x k>

HF F@kDG@kDk x k

MPMAFB%, RA,:All,:F@kD 12p

f@xD -k x x, G@kD 12p

g@xD-k x x>>,

MPCombInt,8All, RV

8x, x

, x


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MPAFEBHF -

1

x2 +a2 x, MPTransInt,

8All, VT 8x, q, x a Tan@qD


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MPChSumIntervalBHF n=-

an,8n,81, 2


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Expansion of A

MPExpandVecA A, AE

I A

M

- 2 A

Deleting a column from a matrix

Table@i j,8i, 3


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MPVecToCart@80, 0, 1F

y@xD+ y@xD

g@sD+g

@s

D2 sA case of two variables

x,x f@x, yD- y,y f@x, yDMPTransEqB%, f@x, yD g@x, hD,

VT :8x, y, Apply SimplifyF

- fH0,2L@x, yD+ fH2,0L@x, yD

gH1,1

L@x, hD Abbreviation of derivatives

MPAbbrevFApH0,1L@x, tD -2 a pH1,0L@x, tDqH1,0L@x, tD, Deriv TrueEpt- 2 a px q x

The subscripts denote the derivatives. The function arguments can be restored using MPRe-

storeFunctions.

MPRestoreF@pt - 2 a pxqx,8p, q


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MPInsideFunc@f@a xD+ a A, CurlH, aD Ha AL+ f@a xD

Merging arguments of a linear function

MPMergeFunc@a f@x, zD+ b f@y, zD, fDf@a x+ b y, zD

Operator Analysis

Expansion of operator expressions

MPExpandBraket@HXa + Xb LAH a\ + b\L, AD

Xa A a\+ Xa A b\+ Xb A a\+ Xb A b\

MPCommutatorB Ql0 2

+ l0

2

P,Q

l0 2

- l0

2

PFMPExpandOp@%,8Q, P


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The terminal velocity, v0, can be found from the equation of motion as t ; when there is

no acceleration, v

= 0, so

m v

m g- b v 2

MPMAFA%

, RA,9All,9v

0, v

v0==,MPSolve,8All, v0, - 1,MPSepVars,8All, t, Times, Side Right


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vi

v 1

g- b v2

m

v 1 t

v g m

bTanhB b g

mt +ArcTanhB b

g mviFF

v g m

b

vi+ g m

bTanhB b g

mtF

g m

b+vi TanhB b g

mtF

v

v0Ivi+ v0 TanhA tTEM

v0+ vi TanhA tTE

where

:v0 g mb

, T v0

g>

Verify that our solution satisfies the originial equation of motion:

MPAFEBv, MPExpandD,:All, v v0Ivi + v0TanhAt

TEM

v0 + viTanhA tTE

, OverDot t>, Apply SimplifyF

MPMAFB%, MPTrigConvert,8At@2D, Tanh, Apply Simplify,

RR,:At@2D,:v0 g mb

, T v0

g>>, Apply ExpandF

v

v0Iv02 - vi2M

TIv0 CoshA tTE +viSinhA t

TEM2

v

- v2 +v0

2

T v0

v

g-b v2

m

that is, Newton's equation of motion.

Substituting the numerical data, v0and Tare



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:v0 g mb

, T v0

g>

MPMAFB%, MPEli,8All, v0, 1, Keep True>, Apply PowerExpand,

Convert,:At@1, 2D, MileHour

>F

:v0 g mb

, T v0

g>

:v0 0.989949 MeterSecond

, T 0.101015 Second>

:v0 2.21445 Mile

Hour, T 0.101015 Second>

Plot vHtL.



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v g m

b

vi + g m

bTanhB b g

mtF

g m

b+viTanhB b g

mtF

MPMAFB%, RA,:At@2D,:g 9.8 MeterSecond2

, b 700Kilo Gram

Meter,

m 70 Kilo Gram, vi 60Mile

Hour, t t Second>>, Apply PowerExpand,

MPDivFrac,:At@2D, MileHour

>,

Convert,:: Hour MeterMile Second

, 1>,:At@2D, MileHour

>>,

MPDivEq,:All, MileHour

>,Plot,:At@2D,8t, 0, 0.1, Take LastF

v g m

b

vi+ g m

bTanhB b g

mtF

g m

b+vi TanhB b g

mtF

Hour v

Mile

2.21445H60 +2.21445 [email protected] tDL

2.21445+ 60 [email protected] tD

0.00 0.02 0.04 0.06 0.08 0.10tHSecondL

10

20

30

40

50

60

vMile

Hour



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Series Solution of Differential Equations

We apply the method of series soution to the linear (classical) oscillator equation.

(9.84)

2 y

x2+ w2 y 0

with known solutions y = sinHwxL, cosHwxL.

We try the series solution of the form

(9.85)

y

@x

D xk

Ia0 + a1 x +a2x

2 +a3x3 +

M HF

l=0

alxk+l

y@xD xk I +a0+ x a1+ x2 a2+ x3 a3M l=0

al xk+l

with a0 0 and the exponent kand all the coefficients alstill undetermined.

By substituting (9.85) into Eq. (9.84), we have

(9.86)

2 y

x2+ w2 y 0

MPMAFB%, RA,:At@1D, y l=0

alxk+l>, Apply MPEvalDF

yw2 +2 y

x2 0

w2 l=0

xk+l al+l=0

x-2+k+l H- 1 +k + lL Hk + lLal 0

which gives



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w2 l=0

xk+l al+l=0

x-2+k+l H- 1 +k+ lL Hk+ lLal 0MPMAFA%, MPShiftSum,9AtA1, x-2+k+lE,8l, 2


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y Sin@xwDa0

w

From (9.91) and (9.94), the general solution can be put

y c1 Cos@xwD +c2Sin@xwD



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MathematicaLanguage

Language Overview (MathematicaGuide)

Structure of MathematicaExpressions

Expressions (MathematicaTutorial)

Constants

Numbers (MathematicaOverview)

Types of Numbers (MathematicaTutorial)

Integer (Built-in MathematicaSymbol)

Rational (Built-in MathematicaSymbol)

Real (Built-in MathematicaSymbol)

Complex (Built-in MathematicaSymbol)

Mathematical Constants (MathematicaGuide)

Variables

Defining Variables (MathematicaTutorial)

Eliminating Variables (MathematicaTutorial)

Patterns

Patterns (MathematicaGuide)

Introduction to Patterns

Verbatim Patterns

Example

In four-wave mixing, two waves of frequencies w1and w2interact and generate additional

frequencies through nonlinear mixing. Suppose the two waves are given by



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Ei 1

2IEi - wi t + Ei* witM

for i = 1, 2, and the nonlinear process is due to the third-order Kerr effect:

PNL e0 c3 E3

Find out what frequencies are present in the output wave near w1 w2.

The field is given by

E HF i=1

2

Ei

MPMAFB%, RA,:At@2D, Ei 12

IEi - wit + Ei* wi tM>,MPEvalSum, At@2DF

E i=1

2

Ei

E 1

2i=1

2

I-t wi Ei+ t wi HEiL*M

E 1

2I-t w1 E1+ -t w2 E2+ t w1 HE1L* + t w2 HE2L*M

and the nonlinear mixing produces

PNL e0 c3 E3

MPMAFB%, RA,:At@2D, E 12

I-t w1 E1 + -t w2 E2 + t w1 HE1L* + t w2 HE2L*M>,Expand, At@2, ED, FactorExp tF

PNL E3 e0 c 3

PNL 1

8e0 c 3I-t w1 E1+ -t w2 E2+ t w1 HE1L* + t w2 HE2L*M3

PNL

1

8 e0 c 3J-3 t w1

E

1

3

+3

t

H-2w1-w2

L E

1

2

E

2+ 3

t

H-w1-2w2

L E

1 E

2

2

+

-3 t w2

E

2

3

+3

-t w1

E

1

2

HE

1L*

+

6-t w2 E

1 E

2HE1L* +3 tHw1-2w2L E22 HE1L* +3 t w1 E1IHE1L*M2 +3 tH2w1 -w2L E2IHE1L*M2 +3 t w1 IHE1L*M3 + 3tH-2w1+w2L E12 HE2L* +6 -t w1 E1 E2HE2L* +3 -t w2 E22 HE2L* +6 t w2 E

1HE1L* HE2L* +6 t w1 E2HE1L* HE2L* +3 tH2w1+w2LIHE1L*M2 HE2L* +

3 tH-w1 +2w2L E1IHE2L*M2 +3 t w2 E2IHE2L*M2 + 3tHw1+2w2LHE1L* IHE2L*M2 + 3 t w2 IHE2L*M3N

The frequencies are



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PNL 1

8e0 c3J-3 t w1 E13 +3 tH-2w1-w2L E12 E2 + 3 tH-w1-2 w2L E1 E22 + -3 t w2 E23 +3 -t w1 E12 HE1L* +

6-t w2 E

1 E

2HE1L* +3 tHw1-2w2L E22 HE1L* +3 t w1 E1IHE1L*M2 + 3tH2w1 -w2L E2IHE1L*M2 +3 t w1 IHE1L*M3 + 3tH-2w1 +w2L E12 HE2L* +6 -t w1 E1 E2HE2L* +3 -t w2 E22 HE2L* +6t w2 E1HE1L* HE2L* +6 t w1 E2HE1L* HE2L* +3 tH2w1+w2LIHE1L*M2 HE2L* +3tH-w1 +2 w2L E1IHE2L*M2 +3 t w2 E2IHE2L*M2 + 3tHw1+2w2LHE1L* IHE2L*M2 + 3 t w2 IHE2L*M3N

MPMAFA%, frequencies Cases@, a_,80,


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f@100D93326215443944 152 681 699 238 856 266 700 490 715 968 264 381 621 468 592 963 895 217 599 993 229

915 608 941 463 976 156 518 286 253 697 920 827 223 758 251 185 210 916 864 000 000 000 000 000 000

000000

100 !

93326215443944 152 681 699 238 856 266 700 490 715 968 264 381 621 468 592 963 895 217 599 993 229

915 608 941 463 976 156 518 286 253 697 920 827 223 758 251 185 210 916 864 000 000 000 000 000 000

000000

Clear@fD

Functional Programming (MathematicaGuide)

Pure Functions (MathematicaTutorial)

Functional Operations (MathematicaOverview)

Example: data manipulation

Generate the data

data = RandomReal@8- 10, 10


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Select@data, NegativeD8- 2.71293, - 7.84716, - 2.51371, - 7.20865, - 0.35902, - 5.73894, - 1.13296, - 9.28741,

- 3.3001, - 6.56867, - 4.45742, - 3.50691, - 5.22019, - 7.31909, - 1.75738, - 1.42342,

- 1.08151, - 8.43983, - 1.63389, - 5.64446, - 6.33726, - 5.05503, - 6.70792, - 1.25488,

- 6.64355, - 4.31986, - 7.33067, - 5.96476, - 9.22198, - 4.24909, - 6.01423, - 0.97207,- 0.425487, - 1.08979, - 4.19288, - 4.14596, - 3.44104, - 5.51746, - 3.39021, - 8.64063,

- 2.70173, - 7.25148, - 9.66792, - 8.06609, - 7.88423, - 6.21864, - 4.81503, - 0.399189

youngjoo chung - tutorial on symbolic computing with mathematica [2011] [p47]

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