Contents. Implementing the QR factorization The algebraic eigenvalue problem. Applied Linear Algebra in Geoscience Using MATLAB

Transcription

1 Applied Linear Algebra in Geoscience Using MATLAB

2 Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in MATLAB User-Defined Functions and Function Files Polynomials, Curve Fitting, and Interpolation Applications in Numerical Analysis Three-Dimensional Plots Symbolic Math Matrices Linear equations Determinants Eigenvalues and eigenvectors Orthogonal vectors and matrices Vector and matrix norms Gaussian elimination and the LU dec. Linear system applications Gram-Schmidt decomposition The singular value decomposition Least-squares problems Linear Independence Basis of a Subspace The Rank of a Matrix Implementing the QR factorization The algebraic eigenvalue problem

3 Background Learn MuPAD by Stare & Compare (examples) Hacking And, if all else fails, then HELP! 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

4 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. 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.

5 Symbolic Computation The Differences Between Symbolic and Numeric Computation Numerical Variables Represent Numbers Answers can Only be Numbers Numeric Computation can be done using Standard Programming Languages Symbolic Variables are Symbols; e.g., x, y, z, β, θ, σ Answers can contain Variables and Functions Symbolic computations are not similar to standard programming languages they use symbolic Manipulations

6 NoteBoook 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

7 NoteBook Regions In MATLAB command window type mupad CALC Regions

8 Capabilities TEXT Regions

9 Capabilities OutPut Regions

10 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 Place the cursor IN the CALC Region Then Hit the Evaluate Button to initiate the Calculation Note also the ASSIGNMENT Operator := After EVALUATE

11 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

12 Syntax & Structure Reserved Symbols PI e E i j 1 I The Logarithms log ln b Base - b log Base - e log A Decimal

13 FLOAT command The float command produces a DECIMAL Result for an otherwise integer display Float(%) Confusion Bottom Expression Evaluated LAST First Expression Edited then Eval d 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

14 Command Bar Shortcuts to Commands in: The General Math Menu Expand Simplify Factor Combine Normalize Rewrite Evaluate Solve Items on the Command bar Derivatives Limits Sums Integrals Rewrite Expressions Products Solve Equations Simplify Evaluate with x = a Numerical Evaluation and Rounding Equality Tests Assignment Math Operators Factorials Function Definition Trig Functions Exponentials and Logs Piecewise Definitions Reserved Symbols Greek Letters Physical Units Matrices and Vectors 2D Plot 3D Plot Items on the Command Bar use # as a PlaceHolder that MUST be replaced

15 Expression vs. Function An EXPRESSION Example A MuPAD Function is called a Procedure Construction looks like an anonymous fcn Use -> in place

16 Plot Plot from a Procedure (UDF) Plot from an Expression No Semi-Colon

17 Printing from MuPAD Each NoteBook must be Printed Separately Use standard Windows Pull-Down Menu: File Print

18 Copy & Paste Pasting in MSWord requires TWO Copy-n-Paste Operations 1. Paste TEXT Regions NORMALLY 2. Paste Script Math Using Paste Special Example of the Bitmap into PPT shown below Copying GRAPHS requires at THIRD Operation Click the Graph to Activate the Plot Menu Bar The Thick border indicates graphic has been copied to the clipboard

19 Algebra Given Expressions 2 y 5 y 3 2 A y 15y 75y B 20 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

20 Solve(eqn)

21 Solve Systems of Eqns z y x z y x z y x z y x z y x z y x Note the use of SQUARE Brackets

22 Solve :: numeric Some eqns do NOT have Algebraic solution, 2 such as x 14 7cos x 10 Solutions x = ±2.76 Solve by regular MATLAB fzero 2 x 14 7 cos x Use fplot for X: Y: X: 2.76 Y: ParabCos x.^ *cos(x); fplot(parabcos,[-5 5]), grid sl = fzero(parabcos, -3) sr = fzero(parabcos, +3) RESULT: sl = sr =

23 Solve :: numeric Check with MuPAD Now use numeric modifier in MuPAD x cosx 0 No Symbol Soln Only ONE Solution LEFT Solution RIGHT Solution

24 Capabilities Click the Object Browser Button Explore the options in the two boxes at right (click upper item first) RIGHT Click the Graphic, then Follow the Wizard instructions

25 Derivatives

26 Capabilities Given: u cosh5v sinh(3v) du dv v0.3

27 Integrals

28 ODEs

29 ODEs Solve this ODE 2 d y 2 dx ky 0 The assume command notifies MuPAD about the Constraint With Boundary Conditions y 0 1 dy dx 0 0 x With Constraint k 0