[email protected] ENGR-25_Lec-25_SimuLink-2.ppt 1 Bruce Mayer, PE Engineering/Math/Physics...

[email protected] • ENGR-25_Lec-25_SimuLink-2.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Chp11:

MuPAD



Learning Goals

Understand and Open MuPAD “NoteBooks”

Work within the NoteBooks to perform SYMBOLIC CALCULATIONS including• Algebra• Transcendental Equations• Calculus• Ordinary

Differential Equations



Chapter Sections NOT Covered

§4 → Matrix Operations §7 → LaPlace Transforms §8 → Special Functions

Carefully Study the Book for Syntax• Learn MuPAD by

– “Stare & Compare” (examples)– Hacking– And, if all else fails, then HELP!



Learning Goals

Use MuPAD to create PLOTS of functions



MATLAB Symbolic MATH

In Addition to Using NUMBERS MATLAB can Manipulate SYMBOLS

Basically, a Mathematics HandBook has been codifed in Software by a group at the University of Paderborn in Germany

The MathWORKS Purchased The “MuPAD” Symbolic Math “Engine” (or “Kernal”), and Integrated it into MATLAB• MuPAD uses Pascal-like Syntax



MuPAD → SYMBOLIC Calcs

MuPAD, sort of Like SimuLink, is a program within-a Program• SimuLink creates Icon-based “Models” (.mdl)

of Mathematical Systems– SimuLink produces NUMERICAL Resuls

MuPAD creates “NoteBooks” (.mn) in which SYMBOLIC quantities can be used• The Symbols can be associated with NO

particular number value– It’s similar to a Math Book that does the work



Symbolic Computation The Differences Between Symbolic and

Numeric Computation

Numerical SymbolicVariables Represent

NumbersVariables are Symbols; e.g., x, y, z, β, θ, σ

Answers can Only be Numbers

Answers can contain Variables and Functions

Numeric Computation can be done using

Standard Programming Languages

Symbolic computations are not similar to standard programming languages – they use symbolic Manipulations



MuPAD BackGround

MuPAD is a part a generic class of Software called a “Computer Algebra System” (CAS). • MuPAD research group at the University of

Paderborn, Germany originally developed this symbolic-manipulation software

The MathWORKS purchased MuPAD to replace the previous “Symbolic Math Kernal” written by MapleSoft



Sample of MuPAD Capabilities

Create SYMBOLIC expressions and manipulate them algebraically.

Obtain SYMBOLIC and numeric solutions to algebraic and transcendental equations.

Perform SYMBOLIC LINEAR ALGEBRA operations, including obtaining expressions for determinants, matrix inverses, and others.



Sample of MuPAD Capabilities

Perform SYMBOLIC differentiation and integration

Evaluate limits and series SYMBOLICALLY.

Obtain SYMBOLIC solutions to Ordinary Differential Equations (ODEs).

Solve ORDINARY DIFFERENTIAL EQUATIONS in terms of special functions or series.



MuPAD “NoteBook”

MuPAD creates within MATLAB its own working SubWindows Called NoteBooks• The User works/types in the “NoteBooks”• NoteBooks saved with extension: .mn

For MuPAD beginers it’s best start at the MuPAD “Welcome” screen

At the MATLAB Command Prompt type mupadwelcome



mupadwelcome Screen

Suggest Spending Some time in the “Getting Started” section



Go Directly to WorkBook In MATLAB

command window type mupad

A MuPAD NoteBook



NoteBook Regions Calcs

CALC Regions



NoteBook Regions Text

TEXT Regions



NoteBook Regions OutPut

OutPut Regions



Importance of “Evaluation”

The CALCULATION regions in MuPAD can be Inserted, Deleted, Edited

After Editing, We may have to EVALUATE the edit using the TOOL or PULL DOWN MENU

If a change in an OUTPUT Region does NOT happen as expected; then be SURE to EVALUATE the CALC region• “EVALUATE ALL” often works best



Evaluate by TOOL Place the cursor

IN the CALC Region

Then Hit the Evaluate Button to initiate the Calculation

Note also the ASSIGNMENT Operator →

After EVALUATE:=



Evaluate Options Use the Pull-

Down Menu for Different forms of the EVALUATION

Use the NOTEBOOK Pull-Down • “Evalute All”

works well



MuPAD Syntax & Structure TEXT regions do

NOT have a left Bracket

CALC & OUTPUT Regions DO have a Left Bracket

Interaction with INTEGERS Produces a SYMBOLIC OutPut

Using a decimal Point Produces a NUMBER Result



MuPAD Syntax & Structure Reserved Symbols

The Logarithms

Log Examples

I1

E

PI

ji

e

log e-Baseln

log b-Baselog

b

A Decimal



FLOAT command The float

command produces a DECIMAL Result for an otherwise integer display

Caveat on float(%) The float(%)

command changes the LAST EVALUATED result to a decimal; not necessarily the value right above (or below) the float(%) command



Float(%) Confusion Bottom Expression

Evaluated LAST First Expression

Edited then Eval’d



Command Bar ShortCuts to

Commands in:

The General Math MenuExpand SimplifyFactor CombineNormalize RewriteEvaluate Solve

Items on the Command barDerivatives Limits SumsIntegrals Rewrite Expressions ProductsSolve Equations Simplify Evaluate with x = aNumerical Evaluation and Rounding Equality Tests AssignmentMath Operators Factorials Function DefinitionTrig Functions Exponentials and Logs Piecewise DefinitionsReserved Symbols Greek Letters Physical UnitsMatrices and Vectors 2D Plot 3D Plot

Items on the Command Bar use# as a PlaceHolder that MUST be replaced



Expression vs. Function An EXPRESSION

Example A MuPAD Function is

called a Procedure• Construction looks like

an anonymous fcn– Use “->” in place of @



Plots are Easy Plot from a

Procedure (UDF) Plot from an

Expression

No SemiColon



Printing from MuPAD Each NoteBook must

be Printed Separately Use standard Windows

Pull-Down Menu:• File → Print

These should appear in all Notebks• Student Name• ENGR25• HW Problem Number• Date Created/Printed



Copy-n-Paste in MuPAD

The TEXT regions can be copied & pasted in the normal fashion

In MSWord The BLUE “Hand Math” must be copied and “Pasted Special” as a “Device Independent Bitmap”

In PPT A Plain “Bitmap” Paste-Special also works

Examples of the two Bitmaps into PPT shown below




Pasting in MSWord requires TWO Copy-n-Paste Operations1. Paste TEXT Regions NORMALLY

2. Paste “Script Math” Using “Paste Special”



Copy-n-Paste in MuPAD Copying GRAPHS

requires at THIRD Operation

Make the Graph

Click the Graph to Activate the Plot Menu Bar



Copy-n-Paste in MuPAD Use Pull-Down

Menu The Thick border

indicates graphic has been copied to the clipboard




Then NORMAL Paste into Word




To AVOID All this Copy-n-Paste Hassle• Simplest is to just PRINT the MuPAD

NoteBook– Only get one Problem per sheet

• Do an alt+PrtScrn to copy the MuPAD window, then paste the whole window into MSWord

• Do an entire SCREEN-SHOT and crop it back to the MuPAD part– If MuPAD window not maximized



Example Algebra

Given Expressions

For This situation find• The Product A∙B in Simplest Form• The Quotient A/B in Simplest Form• Evaluate A+B at y = 8.3 in

– Symbolic form– Numeric form

yyByyyA 2057515 223



Alg

ebra E

xamp

le



Solving Equation solve(eqn)

SOLVE in here



Solve Systems of Eqns Solve by MuPAD

and

Note the use of SQUARE Brackets

35121

214311

111211

zyx

zyx

zyx

364

202

117342

zyx

zyx

zyx



solve::numeric Some eqns do NOT

have Algebraic solution, such as

Solve by regular MATLAB → fzero

Use fplot for

xx cos7142

0cos7142 xx

-5 -4 -3 -2 -1 0 1 2 3 4 5-25

-20

-15

-10

-5

0

5

10

X: -2.76Y: 0.1141

X: 2.76Y: 0.1141

Solutions ≈ x = ±2.76

ParabCos = @(x) x.^2-14 - 7*cos(x);fplot(ParabCos,[-5 5]), gridsL = fzero(ParabCos, -3)sR = fzero(ParabCos, +3)

RESULT:sL = -2.7460sR = 2.7460



solve::numeric Chk with MuPAD

Now use numeric modifier in MuPAD

No Symbol Soln

Only ONE Solution

LEFT Solution

RIGHT Solution



Formatting Plots Make a plot, then

click the plot to CHANGE the tool Bar

Click the “Object Browser” Button

Explore the options in the two boxes at right (click upper item first)



After some menu Exploration



EXPORT Graphs

RIGHT Click the Graphic, then Follow the “Wizard” instructions



Graph Export (JPEG) Result



Derivatives diff(f,x)



Example diff at a value A chance to use the subs command

Given:

Find by MuPAD

)3sinh(5cosh vvu

3.0vdv

du



Integrals int(f,x)



ODEs solve(ode(#, #(#)))



Example assume Solve this ODE

With Boundary Conditions

With Constraint

The assume command notifies MuPAD about the Constraint

02

2

kydx

yd

010 0 xdxdyy

0k



All Done for Today

MuPADused tobe Free

A 523 Page MuPAD Tutorialhttp://www.calvin.edu/~tmk5/research/mupad_tutorial.pdf



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Appendix

6972 23 xxxxf

tfyyy

735



Tutorial on Plot Formating Use

ENGR25_MuPAD_tutorial_format_plot_1111.mn

Solve Make Graph Format with

OBJECT BROWSER tool

[email protected] ENGR-25_Lec-25_SimuLink-2.ppt 1 Bruce Mayer, PE Engineering/Math/Physics...

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