Mastering Matlab 7 By Duane C Hanselman, Bruce L. Little Field.pdf 6i2h67

MATLAB introduction

We cannot teach people anything. We can only help them discover it within themselves. Galileo Galilei

Why this Course? The computer: ‡ indispensable tool used in all aspects of the sciences. ‡ perform time consuming tasks quickly. ‡ the single most important instrument you need to PDVWHUDOORZLQJ\RXWR« ± Visualize, explore, and analyze your data ± Develop models to explain your data ± Simulate phenomena ± ... CourseIntroduction_3

Matlab is a Marketable Skill ‡ Check Job DG¶V ‡ Tiobe index

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Goals ‡ Aim: getting to know the basics Matlab ± ± ± ±

Introduce Matlab to the novice Illustrate key features and capabilities of Matlab Understand basic Matlab code Demonstrate to write efficient Matlab code

‡ Matlab programming and C ± Syntax is similar ± language structure is similar to C: ‡ Matlab s variables, arrays, structures, subroutines, files ‡ Matlab does not pointers and does not require variable declarations

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Goals ‡ Matlab is an extensive package, you will not be able to know all functions ± ± ± ±

Matlab documentation: more than 5000 pages More than 300 built-in functions More than 1000 M-files contained in the base product of Matlab ...

‡ Matlab is the SWISS ARMY KNIFE for numerical problems

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www.mathworks.com Statistics Optimization Signal processing Image processing Curve fitting Bioinformatics toolbox Financial toolbox ....

Application development tools: Matlab compiler Matlab Excel builder Matlab Com builder

Data acquisition and access tools Excel link Data acquisition toolbox.... Database toolbox

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Fortran and Scientific Computing ‡ Engineering and scientific applications involve a lot of "number crunching". ‡ For many years, the main language for this was FORTRAN ‡ Here's a Fortran code to solve a x2 + b x + c = 0: C

Solve a quadratic equation (this is a comment). DESC = B*B - 4*A*C IF ( DESC .LT. 0.0 ) GOTO 10 DESC = SQRT(DESC) X1 = (-B + DESC)/(2.0*A) X2 = (-B - DESC)/(2.0*A) WRITE(6,*) "SOLUTIONS ARE ",X1," AND ", X2 RETURN 10 WRITE(6,*) "EQUATION HAS COMPLEX ROOTS" RETURN CourseIntroduction_8

Solving a Linear System in Fortran Here's a Fortran code to solve a linear system A.x=b, solve for x. C Solve B = A*X for X. C N is dimension of vectors and matrix C Does not use row interchange, scaling. SUBROUTINE LINSYS(N, A, X, B, TMP) INTEGER N DOUBLE PRECISION A(N,N), X(N), B(N) DOUBLE PRECISION TMP(N), RATIO C... Forward elimination DO 13 J=1,N-1 DO 12 I=J+1,N RATIO = -A(I,J)/A(J,J) A(I,*) = A(I,*) +RATIO*ROW(J,*) DO 11 K=J+1,N 11 A(I,K) = A(I,K) + RATIO*A(J,K) A(I,J) = 0.0 X(I) = X(I) + RATIO*X(J) 12 CONTINUE 11 CONTINUE Continued...

C... Backwards substitution X(N) = X(N)/A(N,N) DO 21 I=N-1,1,-1 TMP = X(I) DO 20 J=I+1,N 20 TMP = TMP - A(I,J)*X(J) X(I) = TMP/A(I,I) 21 CONTINUE RETURN END

This is just a small example. A full program may be 1000's of lines long.

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Need for Numerical Libraries ‡ The U.S. government recognized these problems, and the inefficiency of many engineers all writing the same algorithms... again and again. ‡ So, they commissioned numerical analysts to write good quality algorithms for common tasks. ‡ Make the results freely available as "libraries" of subroutines than anyone can use in their programs. ‡ Libraries are available at: www.netlib.org

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History LINPACK, EISPACK. (1970¶V ± In the mid-1970s, Cleve Moler and several colleagues developed the FORTRAN subroutine libraries called LINPACK and EISPACK under a grant from the National Science Foundation. ± LINPACK was a collection of FORTRAN subroutines for solving linear equations, while EISPACK contained subroutines for solving C.... eigenvalue problems. ± Together, LINPACK and EISPACK represented state C.... of the art software for matrix C.... computation.

Jack Dongarra, Cleve Moler, Pete Stewart, and Jim Bunch in 1978

factor the A matrix CALL SGEFA(A, N, N, IPVT, INFO) copy B vector into X vector CALL SCOPY(N, B, 1, X, 1) solve the system of equations CALL SGESL(A, N, N, IPVT, X, 0) CourseIntroduction_11

History ‡ Matlab 0 (1978) ± Cleve Moler designed (as a "hobby" on his own time ) it to give his students interactive access to LINPACK and EISPACK without having to learn FORTRAN ± Moler named his program Matlab, for MATrix LABoratory. ± Over the next several years, when he would visit another university to give a talk, or as a visiting professor, he would end up by leaving a copy of his Matlab on the university machines. ± Within a year or two, Matlab started to catch on by word of mouth within the applied math community as a "cult" phenomena

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History ‡ Matlab 1. (1984) ± reprogrammed in C ± commercial potential => Mathworks ± 1983, John Little was exposed to Matlab because of a visit Cleve made to Stanford. ± Little, an engineer, recognized the potential application of Matlab to engineering applications. ± Little teamed up with Cleve Moler and Steve Bangert to develop a second generation, professional version of Matlab written in C and integrated with graphics. ± The MathWorks, Inc. was founded in 1984 to market and continue development of Matlab.

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History ‡ Software has evolved into an interactive system and programming language for general scientific and technical computation and visualization ‡ The MathWorks has become a commercial success. ± In the period 1984 ± 1991 the number of employees has doubled every year, from 2^0 people in 1984 to 2^7 people in 1991. ± In the following years, the staff has increased roughly 20% per year, from 2^7 people in 1991, to 2^9 people in 1999, and 2^10 people in 2002.

‡ Matlab 7 (2004) ± Release 14

‡ Matlab 2009a/b (2009)

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Matlab: pro ‡ Ease of use: interpreter and integrated environment ± easy and fast coding ± simple, compact, and procedural language with moderate learning curve

‡ Strong graphical and numerical capabilities ‡ Platform independent ‡ Predefined functions (toolbox - Spectral Analysis, Image Processing, Signal Processing, Financial, Symbolic Math ...) ‡ Extra functions can be created in M-files. ‡ Large base with much -contributed software ‡ GUI: can construct its own gui ‡ Lots of code and information available on te web CourseIntroduction_15

Matlab: contra ‡ Interpreter can be slow (compiler) ‡ Few data types/structures ed ‡ ³+DQGOH*UDSKLFV´IRUFXVWRPL]LQJGLVSOD\ options is difficult to use. ‡ well written FORTRAN / C code can be faster ‡ not (yet) suitable for parallel programming ‡ :HEE :LOVRQ'U'REE¶V-RXUQDO-DQ1999 ³/LNHHYHU\RWKHUVFULSWLQJODQJXDJH0DWODE began as a simple way to do powerful things, and it has become a not-so-simple way to do YHU\SRZHUIXOWKLQJV´

‡ cost

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Why not C, FORTRAN, ...? ‡ No built-in graphics; problem to visualize data and models ‡ More bookkeeping needed to do things ‡ pro: Open-source versions exist, some better suited for specific applications

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What about Excel? Spreadsheet programs are very good at dealing with table data in simple ways, and has graphics built-in ‡ More advanced calculations require programming in Visual Basic ‡ Advanced mathematics? ‡ Proprietary, binary file format ‡ Not available on all platforms

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Free Matlab Alternatives ‡ GNU Octave (www.octave.org) is a high-level language, that is mostly compatible with Matlab; it is freely redistributable.

‡ Scilab (www.scilab.org) is a scientific software package for numerical computations providing a powerful open computing environment for engineering and scientific applications dictionary: http://www.scilab.org/product/dic-mat-sci/SCI2M.htm

Using vs Programming Matlab ‡ Some problems can be solved using a software tool (e.g., a spreadsheet program, Excel). It is usually faster to solve a problem with an existing tool. ‡ Many problems cannot be solved with such tools, or a tool is not available. We must then program our own ³WRRO´XVLQJDVXLWDEOHFRPSXWHUODQJXDJH ‡ Matlab blurs the line between computer tool and computer language since it is both.

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References ‡ Engineering Computation with MATLAB David Smith Pearson 2008 ‡ Mastering Matlab 7 Duane Hanselman & Bruce Littlefield Prentice Hall, 2004

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References ‡ use the mathworks website: ± Matlab getting started http://www.mathworks.com/access/helpdesk/help/pdf_doc/matl ab/getstart.pdf ± Matlab reference (3 volumes) http://www.mathworks.com/access/helpdesk/help/pdf_doc/matl ab/refbook.pdf ± http://www.mathworks.com/nn_books ± http://www.mathworks.com/company/newsletters/

Google is your friend

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MATLAB desktop

Introduction to Matlab ‡ Discover how Matlab is designed and what the basic components do ‡ Topics ± Matlab Desktop ± HELP ± Basic language syntax

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Matlab Desktop 7.8

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Current Directory ‡ Location of the folder that contains scripts that you have written and any data files you want to analyze

Matlab Path and Search Order ‡ Matlab finds the functions you call by searching the Matlab Path in the order in which the directories on the path are specified. ‡ If multiple functions with the same name exist on the path, the version associated with the end of the pathlist is used. ‡ The easiest way to add new directories or view the existing path is to use the graphical pathtool available via file->set path or by typing pathtool at the command prompt.

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Workspace/Array editor ‡ Lists variables from code and experiments you have run

‡ double-click the variable opens a window showing the values ‡ values can be changed interactively Desktop_9

Command Window ‡ main working area ‡ can run lines of code ‡ displays values if code is not concluded with ; ‡ pushing up arrow cycles through previous entries

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Command History ‡ displays a log of the statements most recently run in the Command Window ‡ can copy from this list one or more lines of code and paste it into the command window ‡ saves history (up to 20k)

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Desktop layout ‡ Layout can be changed ‡ Different windows can be shown (or not) ‡ Windows can be docked/undocked

HELP ‡ Click on Help-icoon ‡ helpdesk ‡ help + command name: online help in Command Window ‡ helpwin + command name: online help in Help Window ‡ doc + command name: online documentation ‡ lookfor + search string: search for search string in Matlab path

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HELP

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HELP >> help HELP topics matlab\general matlab\ops matlab\lang matlab\elmat matlab\elfun matlab\specfun matlab\matfun matlab\datafun matlab\polyfun matlab\funfun matlab\sparfun matlab\scribe matlab\graph2d matlab\graph3d matlab\specgraph matlab\graphics matlab\uitools matlab\strfun matlab\imagesci matlab\iofun matlab\audiovideo matlab\timefun matlab\datatypes matlab\verctrl matlab\codetools matlab\helptools matlab\winfun matlab\demos

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General purpose commands. Operators and special characters. Programming language constructs. Elementary matrices and matrix manipulation. Elementary math functions. Specialized math functions. Matrix functions - numerical linear algebra. Data analysis and Fourier transforms. Interpolation and polynomials. Function functions and ODE solvers. Sparse matrices. Annotation and Plot Editing. Two dimensional graphs. Three dimensional graphs. Specialized graphs. Handle Graphics. Graphical interface tools. Character strings. Image and scientific data input/output. File input and output. Audio and Video . Time and dates. Data types and structures. Version control. Commands for creating and debugging code. Help commands. Desktop_14 Windows Operating System Interface Files (COM/DDE) Examples and demonstrations.

The Matlab Environment ‡ Matlab is an interpreted language ± Commands are typed into the COMMAND Window and executed immediately ± Variables are allocated in memory as soon as they are first used in an expression ± Commands must be re-entered to be re-executed

‡ All variables created in the Command Window are in what is called the Base Workspace ± Variables can be reassigned new values as needed ± Variables can be selectively cleared from the workspace Desktop_15

Commands, Statements & Variables At the prompt in the Command Window, you can enter either a: ± Command: ‡save mydata (saves workspace in mydata.mat) ‡whos (displays list of workspace variables)

± Assignment Statement: ‡A = width * length; ‡B = 267; ‡ Assignment statements can have only a single variable on the left side of the assignment operator (=) ‡ The RHS is evaluated using current values for all variables and the resulting value is assigned to the variable on the LHS. ‡ Values can be numbers or characters ‡ 7KHYDULDEOH¶V7<3(LVDOZD\VXSGDWHGZKHQHYHUDQHZ assignment is made (warning: powerful but can lead to PLVWDNHV«

± Variables Desktop_17 ‡ up to 63 characters (more are ignored); Case Sensitive ‡ Check with namelengthmax

Editing keys Key

ctrl ctrl home end esc del backspace tab

Function Browse backward through the commands Browse forward through the commands Scroll backward through a single command Scroll forward through a single command Scroll backward through a single command by word Scroll forward through a single command by word Jump to the beginning of the command line Jump to the end of the command line Clear line Discard character under cursor Discard character in front of the cursor Complete a command (select from the list)

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Variables and Names ‡ A variable is a placeholder in memory ‡ Variables contain values ‡ Variable names: ± Are case sensitive: Cost, cost, COST are different ± May contain up to 63 characters (more are ignored) ± Must start with a letter, ± May contain numbers and letters ± 0D\127FRQWDLQSXQFWXDWLRQH[FHSW³B´

‡ How do I view the contents of a variable? ± -XVWW\SHWKHYDULDEOHQDPHZLWKRXWDIROORZLQJ³´ Desktop_19

Reserved Words ‡ Matlab has some special (reserved) words that you may QRWUH XVH« ‡ Use the iskeyword to list all reserved words. for end if while function return elsif case otherwise

switch continue else try catch global persistent break Desktop_20

Case Sensitivity ‡ Matlab is case sensitive ‡ File names ± Matlab running on Windows now gives preference to an exact (case sensitive) name match, but falls back to an inexact (case insensitive) match when no exact match can be found. ± Whenever Matlab 7 detects a potential naming conflict related to case sensitivity, it issues a warning.

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Matlab as a calculator ‡ Always see what is being calculated ‡ Can use variables ‡ %\SXVKLQJDQXSĹDQGGRZQĻDUURZVFDQFDOO previously executed commands ‡ By typing the beginning of command and then u arrow can recall commands that start from the typed symbols ‡ Can copy and paste into documents ‡ Can easily plot graphs

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Built-in functions ‡ Matlab offers a wealth of built-in math functions that can be quite helpful for many computational problems ‡ Elementary Matlab functions (help elfun) ± ± ± ±

Trigonometric functions Exponential functions Complex functions Rounding and remainder functions

‡ Specialized Matlab functions (help specfun) ± Specialized math functions ± Number theoretic functions Desktop_23 ± Coordinate transformations

Elementary math functions Trigonometric. ‡ sin - Sine. ‡ sind - Sine of argument in degrees. ‡ sinh - Hyperbolic sine. ‡ asin - Inverse sine. ‡ asind - Inverse sine, result in degrees. ‡ asinh - Inverse hyperbolic sine. ‡ cos - Cosine. ‡ cosd - Cosine of argument in degrees. ‡ cosh - Hyperbolic cosine. ‡ acos - Inverse cosine. ‡ acosd - Inverse cosine, result in degrees. ‡ acosh - Inverse hyperbolic cosine. ‡ tan - Tangent. ‡ tand - Tangent of argument in degrees. ‡ tanh - Hyperbolic tangent. ‡ atan - Inverse tangent. ‡ atand - Inverse tangent, result in degrees. ‡ atan2 - Four quadrant inverse tangent. ‡ atanh - Inverse hyperbolic tangent.

‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

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sec secd sech asec asecd asech csc cscd csch acsc acscd acsch cot cotd coth acot acotd acoth

- Secant. - Secant of argument in degrees. - Hyperbolic secant. - Inverse secant. - Inverse secant, result in degrees. - Inverse hyperbolic secant. - Cosecant. - Cosecant of argument in degrees. - Hyperbolic cosecant. - Inverse cosecant. - Inverse cosecant, result in degrees. - Inverse hyperbolic cosecant. - Cotangent. - Cotangent of argument in degrees. - Hyperbolic cotangent. - Inverse cotangent. - Inverse cotangent, result in degrees. - Inverse hyperbolic cotangent.

Elementary math functions Exponential. ‡ exp - Exponential. ‡ expm1 - Compute exp(x)-1 accurately. ‡ log - Natural logarithm. ‡ log1p - Compute log(1+x) accurately. ‡ log10 - Common (base 10) logarithm. ‡ log2 - Base 2 logarithm and dissect floating point number. ‡ pow2 - Base 2 power and scale floating point number. ‡ realpow - Power that will error out on complex result. ‡ reallog - Natural logarithm of real number. ‡ realsqrt - Square root of number greater than or equal to zero. ‡ sqrt - Square root. ‡ nthroot - Real n-th root of real numbers. ‡ nextpow2 - Next higher power of 2.

Complex. ‡ abs - Absolute value. ‡ angle - Phase angle. ‡ complex - Construct complex data from real and imaginary parts. ‡ conj - Complex conjugate. ‡ imag - Complex imaginary part. ‡ real - Complex real part. ‡ unwrap - Unwrap phase angle. ‡ isreal - True for real array. ‡ lxpair - Sort numbers into complex conjugate pairs. ‡ Rounding and remainder. ‡ fix - Round towards zero. ‡ floor - Round towards minus infinity. ‡ ceil - Round towards plus infinity. ‡ round - Round towards nearest integer. ‡ mod - Modulus (signed remainder after division). ‡ rem - Remainder after division. ‡ sign - Signum.

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Specialized math functions Specialized math functions. ‡ airy - Airy functions. ‡ besselj - Bessel function of the first kind. ‡ bessely - Bessel function of the second kind. ‡ besselh - Bessel functions of the third kind (Hankel function). ‡ besseli - Modified Bessel function of the first kind. ‡ besselk - Modified Bessel function of the second kind. ‡ beta - Beta function. ‡ betainc - Incomplete beta function. ‡ betaln - Logarithm of beta function. ‡ ellipj - Jacobi elliptic functions. ‡ ellipke - Complete elliptic integral. ‡ erf - Error function. ‡ erfc - Complementary error function. ‡ erfcx - Scaled complementary error function. ‡ erfinv - Inverse error function. ‡ expint - Exponential integral function. ‡ gamma - Gamma function. ‡ gammainc - Incomplete gamma function. ‡ gammaln - Logarithm of gamma function. ‡ psi - Psi (polygamma) function. ‡ legendre - Associated Legendre function. ‡ cross - Vector cross product. ‡ dot - Vector dot product.

Number theoretic functions. ‡ factor - Prime factors. ‡ isprime - True for prime numbers. ‡ primes - Generate list of prime numbers. ‡ gcd - Greatest common divisor. ‡ lcm - Least common multiple. ‡ rat - Rational approximation. ‡ rats - Rational output. ‡ perms - All possible permutations. ‡ nchoosek - All combinations of N elements taken K at a time. ‡ factorial - Factorial function. Coordinate transforms. cart2sph - Transform Cartesian to spherical coordinates. ‡ cart2pol - Transform Cartesian to polar coordinates. ‡ pol2cart - Transform polar to Cartesian coordinates. ‡ sph2cart - Transform spherical to Cartesian coordinates. ‡ hsv2rgb - Convert hue-saturation-value colors to redgreen-blue. ‡ rgb2hsv - Convert red-green-blue colors to huesaturationvalue. ‡

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Some Special Variables ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

ans : default variable name for results beep: make sound pi: pi eps: smallest number that can be subtracted from 0 to make a negative number inf: infinity (or Inf) NaN: not a number (or nan) realmin: smallest positive real number realmax: largest positive real number

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Format Format controls display of output in the command window

format format format format

short long rat hex

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Matlab Programs ‡ &DQZHH[HFXWHD³SURJUDP"´RULVLWDVLPSOHFDOFXODWRU" ‡ Programs in Matlab are: ± Scripts: Matlab statements that are fed from a file into the Command Window and executed immediately ± Functions: Program modules that are ed data (arguments) and return a result (i.e., sin(x)) ± These can be created in any text editor (but Matlab supplies a nice built-in editor)

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Editor

Access to commands Color keyed text with auto indents

tabbed sheets for other files being edited Desktop_30

Summary ‡ ‡ ‡ ‡

desktop environment try to use help as much as possible besides Matlab Help there is also google use Matlab interactively as a calculator

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MATLAB numeric data types

Basic Data Structure ‡ array: ± most basic data structure in Matlab ± a two-dimensional, rectangularly shaped data structure ± can contain numbers, characters, logical states

‡ Matlab uses these two-dimensional arrays to store single numbers and linear series of numbers as well. In these cases, the dimensions are 1-by-1 and 1-by-n respectively, where n is the length of the numeric series. ‡ Matlab also s data structures that have more than two dimensions.

data types ‡ Matlab is different from C and other common languages in some important ways: ± basic data structure in Matlab is array. ± variables do not have to be declared beforehand.

overview MATLAB Data Types

double

single

double precision (real and complex)

single precision (real and complex)

int8, int16, int32, int64,

unit8 uint16 unit32 unint64

logical logical data

char character strings

integer and unsigned integer data types

overview

cell

structure

classes

function handles

cell arrays

structures

objects

function handles

Numeric Data Types ‡ Numeric data types in Matlab include: ± signed and unsigned integers, ± single- and double-precision floating-point numbers. ± Integer and single- precision arrays offer more memory efficient storage than double precision.

‡ All numeric types basic array operations, such as subscripting and reshaping. ± All numeric types except for int64 and uint64 can be used in mathematical operations.

integer ‡ 4 signed and 4 unsigned integer data types. ‡ Signed types: ± enable negative integers as well as positive, ± represent smaller range of numbers as the unsigned types

‡ Unsigned types ± wider range of numbers ± zero or positive.

‡ Matlab s 1-, 2-, 4-, and 8-byte storage for integer data. ± save memory and execution time for your programs if you use the smallest integer type that accommodates your data. ± no need for a 32-bit integer to store the value 100.

integer

integer Creating Integer Data ‡ Matlab stores numeric data as double-precision floating point by default. ‡ Store data as an integer: use one of the conversion functions x = int16(32501); ‡ Use the whos function to show the dimensions, byte count, and data type of an array represented by a variable. ‡ ex. ± PAY ATTENTION! x=int8(60); y=int8(70); z=x+y; ‡ ex. demo_integer

floating point: double ‡ Matlab represents floating-point numbers in: ± double-precision (default) ± single-precision format.

‡ single precision with a conversion function (single) ‡ according to IEEE Standard 754 for double precision. 64 bits, formatted as: Bits 63 62 to 52 51 to 0

Usage sign (0 = positive, 1 = negative) Exponent, biased by 1023 Fraction f of the number 1.f

‡ Maximum and Minimum Double-Precision Values: ± realmax and realmin return the maximum and minimum values that you can represent with the double data type ± conversion function: double ‡ ex. demo_float

floating point: single ‡ Matlab constructs the single data type according to IEEE Standard 754 for single precision. Any value stored as a single requires 32 bits: Bits 31 30 to 23 22 to 0

Usage sign (0 = positive, 1 = negative) Exponent, biased by 127 Fraction f of the number 1.f

‡ Maximum and Minimum Double-Precision Values: ± UHDOPD[µVLQJOH¶ and UHDOPLQµVLQJOH¶ return the maximum and minimum values that you can represent with the double data type ± conversion function: single

complex ‡ Complex numbers consist of two separate parts: a real part and an imaginary part. ‡ The basic imaginary unit is equal to the square root of 1. represented by: i or j x = 2 + 3i; ‡ Another way ± use the complex function. x = rand(3) * 5; y = rand(3) * -8; z = complex(x, y)

‡ Separate a complex number into its real and imaginary parts using the real and imag functions: zr = real(z); zi = imag(z);

Infinity and NaN ‡ Infinity ± special value inf. ± results from operations like division by zero and overflow. ± Use the isinf function to that x is positive or negative infinity:

‡ NaN ± values that are not real or complex numbers ± NaN: Not a Number. ± 0/0 and inf/inf result in NaN ± Use the isnan function to that x is NaN: ± Logical Operations on NaN. Because two NaNs are not equal to each other, logical operations involving NaN always return false, except for a test for inequality

x = 1/0 x = Inf x = 1.e1000 x = Inf x = exp(1000) x = Inf x = log(0) x = -Inf

x = log(0); isinf(x) ans = 1 x = 7i/0 x = NaN + Infi x = log(0); isnan(x) ans = 1

MATLAB arrays: creation

ArraysCreation_2

Topics ‡ Creating ± ± ± ± ±

how to build an array in a fast way manual entering the elements : linspace, logspace special functions, special matrices

‡ Indexing ± How to select elements

Help Search for ‡ Creating and Concatenating Matrices ‡ Matrices and Arrays ‡ Matrix Indexing ‡ Elementary matrices

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Matlab Arrays ‡ 0DWODE¶V fundamental data structure is the array Matlab considers each variable to be a matrix and "knows" how big it is. ‡ )XQGDPHQWDORSHUDWRUVHJDGGLWLRQPXOWLSOLFDWLRQ« are programmed to deal with matrices when required. The Matlab environment handles much of the bothersome housekeeping that makes all this possible.

Basic Concepts Scalars: magnitude only

x, mass, color, 13.451 Vectors: magnitude r ANDr direction

force 12.74 i 5.234 j r r r r r H a1i1 a2i2 a3i3 a4i4 L

r anin

Arrays: can be 2D or higher dimension

A

a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34

R

rijk ArraysCreation_5

Creating Arrays ‡ Different ways:

± specify each element explicitly, ± use the colon : operator, ± use the commands linspace or logspace ± elementary (built-in) arrays ‡ The Matlab arrays are indexed as: A(row,column)

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Creating Arrays ‡ Create an array: use the square brackets [ ] ‡ Specify each element explicitly ‡ Numbers (or variables) inside the brackets can be separated by blanks, commas or semicolons ± blank or comma separator: separate elements in a row ± semicolon: separate rows » a = [1 2 3] a = 1 2 3 » b = [11, 2, 3] b = 11 2 » c = [1;2;3] c = 1 2 3

3

ArraysCreation_7

Creating Arrays ‡ always refer to rows first and columns second. 4-by-3 >> A = [1 3 5 ; 2 4 1 ; 3 3 3 ; 2 1 9] A = 1 3 5 2 4 1 3 3 3 2 1 9

‡ create an empty matrix. >> D = [] D = []

Creating Arrays: Colon Operator ‡ Create vectors, array subscripting, and for iterations ‡ The colon (:) is one of the most useful operators in Matlab. from_, to _, in increments of_ ‡ The colon operator uses the following rules to create regularly spaced vectors: ± ± ± ±

j:k is the same as [j,j+1,...,k] j:k is empty if j > k j:i:k is the same as [j,j+i,j+2i, ...,k] j:i:k is empty if i > 0 and j > k or if i < 0 and j < k ArraysCreation_9

Example » D = 1:4 D = 1 2 3 » E = 1:0.3:2 E = 1.0000 1.3000 1.6000 1.9000 » d d = 1 2 3 11 2 3 1 2 3 » d(2,:) ans = 11 2 3 » d(:,1) ans = 1 11 ArraysCreation_10 1

4

» d(:) ans = 1 11 1 2 2 2 3 3 3 » d(2:3) ans = 11 1 » d(8:9) ans = 3 3 » d(end:-1:6) ans = 3 3

3

2

Creating Arrays ‡ linspace ± syntax: x = linspace(first, last, n) ± creates linearly spaced row vector starting with first, ending at last , having n elements

‡ logspace ± syntax: x = linspace(first, last, n) ± creates logarithmically spaced row vector starting with 10first, ending at 10last , having n elements

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Creating Arrays Matlab provides functions for creating standard arrays. >> help elmat Elementary matrices and matrix manipulation. Elementary matrices. zeros - Zeros array. ones - Ones array. eye - Identity matrix. repmat - Replicate and tile array. rand - Uniformly distributed random numbers. randn - Normally distributed random numbers. linspace - Linearly spaced vector. logspace - Logarithmically spaced vector. freqspace - Frequency spacing for frequency response. meshgrid - X and Y arrays for 3-D plots. accumarray - Construct an array with accumulation. : - Regularly spaced vector and index into matrix . ArraysCreation_12

Examples >> ones(3)

>> randn(3,2) ans = -0.4326 0.2877 -1.6656 -1.1465 0.1253 1.1909 >> a=1:4 a =

ans = 1 1 1

1 1 1

1 1 1

>> zeros(2,5)

1 2 >> diag(a) ans =

ans = 0 0

0 0

0 0

0 0

0 0

1 0 0 0

>> eye(3) ans =

0 2 0 0

3

4

0 0 3 0

0 0 0 4

0 0 0 0 3 0

0 0 0 0 0 4

>> diag(a,-2) ans = 1 0 0

0 1 0

0 0 1

0 0 1 0 0 0

>> rand(1,5) ans = 0.9501 0.8913

0.2311

0.6068

0 0 0 2 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0.4860 ArraysCreation_13

Size of a Matrix ‡ determine the size of a matrix by using the size() command >> A = rand(3) A = 0.8147 0.9134 0.2785 0.9058 0.6324 0.5469 0.1270 0.0975 0.9575 >> size(A) ans = 3 3 >> [nrows,ncols] = size(A) nrows = 3 ncols = 3

Number of Elements ‡ numel() command. >> n = numel(A)

n = 9

‡ length() command (vector) ± the statement length(X) is equivalent to max(size(X)) >> length(A) ans = 3

Transposing a Matrix ‡ A m-by-n matrix can be transposed into a n-by-m matrix by using the transpose operator '. >> A = [1 2 3 4 ; 5 6 7 9] A = 1 2 3 4 5 6 7 9 !!% $µ B = 1 5 2 6 3 7 4 9

Concatenating Matrices ‡ Matrices can be concatenated by enclosing them inside of square brackets and using either a space or semicolon to specify the dimension. ‡ Care must be taken that the matrices are of the right size. >> A = [[1 2 3],rand(1,3)] A = 1.0000 2.0000 3.0000 0.9706 >> B = [A;A] B = 1.0000 2.0000 3.0000 0.9706 1.0000 2.0000 3.0000 0.9706

0.9649

0.1576

0.9649

0.1576

0.9649

0.1576

Repeating Arrays ‡ reshape syntax: B = reshape(A,m,n) returns the m-by-n matrix B whose elements are taken columnwise from A. An error results if A does not have m*n elements. ‡ repmat syntax B = repmat(A,m,n) creates a large matrix B consisting of an m-by-n tiling of copies of A. repmat(A,n) creates an n-by-n tiling.

ArraysCreation_18

Reshaping ‡ reshape B = reshape(A,m,n) returns the m-by-n matrix B whose elements are taken columnwise from A. An error results if A does not have m*n elements. ‡

>> x = 1:9 x = 1 5 9

2 6

3 7

4 8

>> reshape(x,3,3) ans = 1 2 3

4 5 6

7 8 9

More functions Function [a,b] [a;b]

or

Description Create a matrix from specified elements, or concatenate matrices together.

accumarray

Construct a matrix using accumulation.

blkdiag

Construct a block diagonal matrix.

cat

Concatenate matrices along the specified dimension.

diag

Create a diagonal matrix from a vector.

horzcat

Concatenate matrices horizontally.

magic

Create a square matrix with rows, columns, and diagonals that add up to the same number.

ones

Create a matrix of all ones.

rand

Create a matrix of uniformly distributed random numbers.

repmat

Create a new matrix by replicating or tiling another.

vertcat

Concatenate two or more matrices vertically.

zeros

Create a matrix of all zeros. ArraysCreation_20

Concatenate cat ‡ Concatenate arrays ‡ Syntax ± C = cat(dim,A,B) ± C = cat(dim,A1,A2,A3,A4...)

‡ C = cat(dim,A,B) concatenates the arrays A and B along dim. ‡ C = cat(dim,A1,A2,A3,A4,...) concatenates all the input arrays (A1, A2, A3, A4, and so on) along dim. ‡ cat(2,A,B) is the same as [A,B] ‡ cat(1,A,B) is the same as [A;B]

Indexing ‡ Extract individual entries by specifying the indices inside round brackets (). ‡ extract several entries at once by specifying ± a matrix, ± use the : operator to extract all entries along a certain dimension.

Indexing: example A = rand(5) A = 0.9572 0.9157 0.4854 0.7922 0.8003 0.9595 0.1419 0.6557 0.4218 0.0357 >> A1=A(3,5) A1 = 0.0971 >> A2 = A([1, 3], 5) A2 = 0.2769 0.0971 >> A3 = A(:,1) A3 = 0.9572 0.4854 0.8003 0.1419 0.4218

0.8491 0.9340 0.6787 0.7577 0.7431

0.3922 0.6555 0.1712 0.7060 0.0318

0.2769 0.0462 0.0971 0.8235 0.6948

Assignment of elements ‡ Assignment operations follows the same rules as indexing and then specify the new values on the right hand side. ‡ The right must be either a scalar value, or a matrix with the same dimensions as the resulting indexed matrix on the left. ‡ Matlab automatically expands scalar values on the right to the correct size

>> A = ones(3,5) A = 1 1 1 1 1 1 1 1 1 >> A(3,2) = 5 A = 1 1 1 1 1 1 1 5 1 >> A(:,1:3:end) = 8 A = 8 1 1 8 1 1 8 5 1

1 1 1

1 1 1

1 1 1

1 1 1

8 8 8

1 1 1

Deletion ‡ Asg [] deletes the corresponding entries from the matrix. ‡ Only deletions that result in a rectangular matrix are allowed. A = 8 1 1 8 1 8 1 1 8 1 8 5 1 8 1 >> A(2,1)=[] ??? Subscripted assignment dimension mismatch. >> A(2,:)=[] A = 8 1 1 8 1 8 5 1 8 1

Expansion ‡ When the indices in an assignment operation exceed the size of the matrix, the matrix will be expanded ‡ No error message! ‡ Be careful! >> A = magic(3) A = 8 1 3 5 4 9 >> A(5,7) = 99 A = 8 1 3 5 4 9 0 0 0 0

6 7 2

6 7 2 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 99

Summary ‡ ‡ ‡ ‡

manual entering the elements : linspace, logspace special functions, special matrices

‡ ex.: demo_array_creation

ArraysCreation_28

MATLAB array operations

Topics ‡ ‡ ‡ ‡ ‡

Mathematical operations Manipulation Sorting and searching More built-in manipulation Multidimensional arrays

ArraysOperations_2

Help Search for ‡ Arithmetic operators ‡ Array operators

Basic Matrix Operations

ArraysOperations_4

Dot (Array) Operations

ArraysOperations_5

Scalar-Array Mathematics ‡ Addition, subtraction, multiplication, and division of an array by a scalar applies the operation to all elements of the array.

>> g = [1 2 3 4; 5 6 7 8] g = 1 5

2 6

3 7

4 8

0 4

1 5

2 6

3 11

5 13

7 15

>> g-2 ans = -1 3 >> 2*g-1 ans = 1 9 >> 3*g/5+4 ans = 4.6000 7.0000

ArraysOperations_6

5.2000 7.6000

5.8000 8.2000

6.4000 8.8000

Array-Array Mathematics ‡ Mathematical operations between arrays are not so simple as those between scalars and arrays ± Addition/subtraction: C=A+B where cij = aij+bij ± Multiplication/division: C=A .* B where cij = aij*bij ± Exponentiation: C=A .^ x where cij = aijx

‡ Same order of precedence rules as under basic scalar operations

ArraysOperations_7

Array-Array Mathematics ‡ Arithmetic operations on arrays are just like the same operations for scalars but they are carried out on an element-by-element basis. ± the dot (.) before the operator indicates an array operator; it is needed only if the meaning cannot be automatically inferred. ± when combining arrays, make sure they all have the same dimensions ± applies to vectors, 2D arrays, multi-dimensional arrays

ArraysOperations_8

Example >> g

>> g + h

g =

ans = 1 5 9

2 6 10

3 7 11

4 8 12

2 7 12

>> h

>> 2*g-h

h =

ans = 1 2 3

1 2 3

1 2 3

1 2 3

1 8 15

3 8 13

4 9 14

5 10 15

3 10 17

5 12 19

7 14 21

38 44 50 56

38 44 50 56

>> ArraysOperations_9

Example >> g*h ??? Error using ==> mtimes Inner matrix dimensions must agree.

» g * h' ans = 10 26 42

>> g.*h

20 52 84

30 78 126

» g' * h ans = ans =

1 10 27

2 12 30

ArraysOperations_10

3 14 33

4 16 36

38 44 50 56

38 44 50 56

More on Array Operations ‡ Most Matlab functions will >> A=[1 2 3 4 5]; >> sin(A) work equally well with ans = both scalars and arrays 0.8415 0.9093 (of any dimension) 0.1411 -0.7568 -0.9589 >> sqrt(A) ans = 1.0000 1.7321 2.2361

1.4142 2.0000

ArraysOperations_11

>> A = [1:3;4:6;7:9] A = 1 2 3 4 5 6 7 8 9 >> mean(A) ans = 4 5

6

>> sum(A) ans = 12

18

ArraysOperations_12

15

Most common functions operate on columns by default

Reversing ‡ reversing rows and columns

>> a = [1 2 3 ; 4 5 6 ; 7 8 9] a = 1 4 7

2 5 8

3 6 9

>> b = a(end:-1:1) b = 9 2

6 7

3 4

8

5

1

>> b = a(end:-1:1,1:3) b =

ArraysOperations_13

7 4 1

8 5 2

9 6 3

Sorting ‡ Sort array elements in ascending or descending order ‡ Syntax ± ± ± ±

B = sort(A) B = sort(A,dim) B = sort(...,mode) [B,IX] = sort(...)

‡ B = sort(A) sorts the elements along different dimensions of an array, and arranges those elements in ascending order. ‡ If A is a ...sort(A) ... ± Vector: Sorts the elements of A. ± Matrix: Sorts each column of ArraysOperations_14 A.

>> a = randperm(8) a = 8 2 7 5 1 >> b = sort(a) b = 1 2 3 7 8 >> A A = 3 7 5 0 4 2 >> sort(A) ans = 0 4 2 3 7 5 >> sort(A,1) ans = 0 4 3 7 >> sort(A,2) ans = 3 5 0 2

2 5

7 4

4

3

6

4

5

6

Sorting a =

‡ B = sort(A,dim) sorts the elements along the dimension of A specified by a scalar dim. If dim is a vector, sort works iteratively on the specified dimensions. Thus, sort(A,[1 2]) is equivalent to sort(sort(A,2),1). ‡ B = sort(...,mode) sorts the elements in the specified direction, depending on the value of mode.

7

6

8

4

2

1

3

5

>> [xs indx] = sort(a, 'descend')

xs = 8 4

7 3

6 2

5 1

indx = 2 6

1 4

5 3

8 7

± DVFHQGµ Ascending order (default) ± GHVFHQGµDescending order

ArraysOperations_15

Searching ‡ The find FRPPDQG³ILQGV´ of an array that meet a criteria. The result of the command is a list of element numbers.

ArraysOperations_16

>> grades = -5: 10: 105 grades = -5 5 15 25 35 45 55 65 75 85 95 105 >> set1 = find(grades>100 | grades <0) set1 = 1 12 >> set2 = find(grades>=0 & grades <=100) set2 = 2 3 4 5 6 7 8 9 10 11 >> grades(set1) ans = -5 105 >> grades(set2) ans = 5 15 25 35 45 55 65 75 85 95

Searching ‡ The find FRPPDQG³ILQGV´ works also in 2 dimensions ‡ more specific ‡ find(X, n): start at the first element returns up to n indices where X is true ‡ ILQG;Q¶ODVW¶ start at the last element returns up to n indices where X is true

ArraysOperations_17

>> a = 1:9 a = 1 2 3 4 7 8 9 >> a = reshape(a,3,3)' a = 1 2 3 4 5 6 7 8 9 >> [ii,ij]=find(a>5) ii = 3 3 2 3 ij = 1 2 3 3 >> k = find(a>5) k = 3 6 8 9 >> l = find(a>5, 2, 'last') l = 8 9

Built-in functions help elmat: Matrix manipulation. ‡ fliplr: Flip matrix in left/right direction. ‡ flipud: Flip matrix in up/down direction. ‡ rot90: Rotate matrix 90 degrees. rot90(a,n): Rotate n-times ‡ circshift(A,shiftsize) circularly shifts the values in the array, A, by shiftsize elements. ± shiftsize is a vector of integer scalars where the n-th element specifies the shift amount for the n-th dimension of array A. ± positive shiftsize: shift down (or to the right). ± negative shiftsize: shift up (or to the left). ± 0: no shift ArraysOperations_18

a = 1 2 3 4 5 6 7 8 9 >> flipud(a) ans = 7 8 9 4 5 6 1 2 3 >> fliplr(a) ans = 3 2 1 6 5 4 9 8 7 >> rot90(a) ans = 3 6 9 2 5 8 1 4 7 >> rot90(a,2) ans = 9 8 7 6 5 4 3 2 1 >> circshift(a,[-1 1]) ans = 6 4 5 9 7 8 3 1 2

5

6

Built-in functions ‡ diag - Diagonal matrices and diagonals of matrix. ‡ tril - Extract lower triangular part. ‡ triu - Extract upper triangular part.

a = 1 2 4 5 7 8 >> diag(a) ans = 1 5 9 >> diag(ans) ans = 1 0 0 5 0 0 >> triu(a) ans = 1 2 0 5 0 0 >> tril(a) ans = 1 0 4 5 7 8

3 6 9

0 0 9

3 6 9

0 0 9

ArraysOperations_19

Multidimensional Arrays ‡ arrays with more than 2 subscripts ‡ Examples: ± 3D physical data ± sequence of matrices ± samples of a time-dependent 2D or 3D data

‡ To make multidimensional array: >> a = [2 4 6; 7 8 9; 1 2 3] >> a(:,:,2)= [1 11 12; 0 1 2; 4 5 6] When you add elements and expand the size Unspecified elements are set to zero >> a(:,:,4) = [ 1 1 1; 2 2 2; 3 3 3]

Summary ‡ mathematics: matrix versus arrays ‡ sorting and searching functions ‡ multidimensional arrays

ArraysOperations_21

Functions & Arrays

ArraysOperations_22

Functions & Arrays

ArraysOperations_23

Functions & Arrays

ArraysOperations_24

Functions on matrix

ArraysOperations_25

MATLAB relational and logical operators

Topics ‡ Use relational operators to test two values ‡ Work with values of true and false ‡ Compare relationships using logical operators ‡ Analyze the precedence of operators

LogicalRelational_2

Help Search for ‡ Logical operator ‡ Relational operator

Logical Expressions Why? ‡ What if you want to choose between two (or more) possible programming paths? ‡ Use a logical condition to decide what to do next ‡ Requires logical operators

LogicalRelational_4

Recall: double data type ‡ Contains numeric data. ‡ Arithmetic operators: ± +, - , *, .*, / , ./, ^, .^

‡ Rules of precedence for arithmetic operators: ± 1. Parentheses ± 2. Exponents ± 3. Multiplication / Division ± 4. Addition/Subtraction ‡ combine arithmetic operators with parentheses as QHHGHGWRFUHDWHPRUHFRPSOLFDWHGH[SUHVVLRQV«

LogicalRelational_5

Relational Operators ‡ Used to compare two numeric values ‡ Returns a value of true or false. ‡ In Matlab, ± 1 = true (any non-zero number); ± 0 = false; ± Logical data type

Relational Operators: < less than <= less than or equal to > greater than >= greater than or equal to == equals to ~ = not equal to

LogicalRelational_6

Operator Precedence 1. Parentheses () 2. Transpose (.'), power (.^), complex conjugate transpose ('), matrix power (^) 3. Unary plus (+), unary minus (-), logical negation (~) 4. Multiplication (.*), right division (./), left division (.\), matrix multiplication (*), matrix right division (/), matrix left division (\) 5. Addition (+), subtraction (-) 6. Colon operator (:) 7. Less than (<), less than or equal to (<=), greater than (>), greater than or equal to (>=), equal to (==), not equal to (~=) 8. Element-wise AND (&) 9. Element-wise OR (|) 10. Short-circuit AND (&&) 11. Short-circuit OR (||)

Operator Precedence ‡ When relational operators are present: ± All arithmetic operations are performed first (in their particular order) ± Then the relational operators are evaluated.

‡ Example 1 (2*3) > (4+1);

- The multiplication and addition are first: 6 > 5

- The relational operator is evaluated: 6 is greater than 5, so this returns 1 (true)

- Result is the numeric value, 1, is returned

LogicalRelational_8

Relational Operators: array ‡ The Matlab relational operators compare corresponding elements of arrays with equal dimensions. ‡ Relational operators always operate elementby-element. ‡ example A = [2 7 6;9 0 5;3 0.5 6]; B = [8 7 0;3 2 5;4 -1 7]; A == B ans = 0 1 0 0 0 1 0 0 0

Relational Operators ‡ vectors and rectangular arrays, both operands must be the same size unless one is a scalar. ‡ one operand is a scalar and the other is not, Matlab tests the scalar against every element of the other operand. ‡ result: ± relation is true receive logical 1 ± relation is false receive logical 0. ± on array manipulations: result is a logical array

LogicalRelational_10

Logical Operators ‡ Logical Operators: ± Provide a way to combine results from Relational Expressions or between logical values ± Returns a value of true or false.

‡ Evaluated after all other operators have been performed.

LogicalRelational_11

Logical Operators & AND | OR ~ NOT xor XOR

Logical Operators ‡ AND: & ± Returns true if two expressions being compared are true. ± Returns false if any of the two is false.

‡ OR: | ± Returns true if any of the two expressions is true. ± Returns false only if the two are both false.

‡ NOT: ~ ± Returns true if the single expression is false. ± Returns false if the single expression is true.

LogicalRelational_12

Examples: ‡ Assume: a=7; b=4; c=3; ‡ ~(a==3*b) ± ± ± ±

Evaluates: 3*b = 12 Evaluates: (a==12) and result is false Evaluates ~(false) and result is true Returns ans = 1 (true)

‡ a > 5 & b > 5 ± Evaluates (a>5) and (b>5) separately. ± One returns true, the other returns false. ± Since both are not true, the expression returns false.

‡ a == 7 | b ==1 ± Evaluates (a==7) and (b==1) separately ± One returns true and the other returns false ± Since at least one is true, the expression returns true

Logical Operators: more 3 types of logical operators and functions: ± Element-wise ² operate on corresponding elements of logical arrays. ± Bit-wise ² operate on corresponding bits of integer values or arrays. ± Short-circuit ² operate on scalar, logical expressions.

‡ logical operators work also on arrays

LogicalRelational_14

Logical Operators: more Element-Wise Operators and Functions

The following logical operators and functions perform element-wise logical operations on their inputs to produce a like-sized output array. examples A = [0 1 1 0 1]; B = [1 1 0 0 1];

LogicalRelational_15

Logical Operators: more Bit-Wise Functions ‡ bit-wise logical operations on nonnegative integer inputs. ‡ inputs may be scalar or in arrays. ‡ examples A % B %

= 28; binary 11100 = 21; binary 10101

LogicalRelational_16

Logical Operators: more Short-Circuit Operators ‡ The following operators perform AND and OR operations on logical expressions containing scalar values. ‡ They are short-circuit operators in that they evaluate their second operand only when the result is not fully determined by the first operand.

LogicalRelational_17

Example example: logical_ex_1.m x=5; y=10; disp(x 10) & (y<30)) disp((x>2) | (y > 89))

LogicalRelational_18

A Common Mistake ‡ You will not get into trouble if you make sure that Logical Operators are always used with logical values. ‡ A > B & C (where A=10, B=5, C=0) ± This looks like a relational expression asking if A is greater than both B and C which should be true for these values. ± Here is what really happens: ‡ A>B is evaluated as true ‡ result (true) is logically ANDed with C ‡ Since Matlab treats any zero numeric as false, it will mistakenly treat C as a logical and the result will be false

± The CORRECT form is: (A > B) & (A > C)and this returns a true result. ± NOTE: subtle problems can arise because Matlab uses numerics to represent logical values. Other languages are stricter in this area DQGZRQ¶WOHWWKLVKDSSHQ

Logical Values in Assignments ‡ True/False values can be assigned to variables DQGWKHQWUHDWHGQXPHULFDOO\LQ0DWODE« ‡ The variables will be assigned the value that is returned from relational and/or logical operators. ‡ The variables will thus have a value of 1 or 0. Example: a=7; b=4; c=3; ± x = a > 2; ‡ Then x = 1;

± y = b==5; ‡ Y will be equal to 0.

‡ This can be very useful in Matlab but it is actually quite DANGEROUS

More Examples a=6; b=10; c=-2; Try the following examples without the use of Matlab: X1 = abs(c)>3 & c<(b-a) & b+a > 3 X2 = (b==10) | (a< 4) X3 = a.*5 >= b.*3 & c < a

Advice: use parentheses to make expression more readable (see example for X2).

LogicalRelational_21

Practice ‡ Evaluate the following: ± ± ± ± ±

a = 4; b = 20; c = 12; d = 5; One = a>4 & b==20 Two = b<40 | c>10 Three = d.*c > a.*b Four = b.*3<= 100 & d<10 & a.*d==b

LogicalRelational_22

More Practice ‡ When comparing vectors, the operator (>, <=, ~, AND, etc.) is applied element-by-element:

a = [0,2,4,2];

b = [4,1,-2,3];

What is: C = a .* b; C = b.^2-a.*b C = a >= b;

‡ Advice: be careful when comparing floating point numbers because the finite precision may cause values to be slightly off LogicalRelational_23

An Interesting Example ‡ Sometimes if you have to do an array division, you may want to avoid getting inf when a value in the divisor is ]HUR«

>> B=[2:-1:-2] B =

2

1

0

-1

-2

>> numerator=ones(size(B)) numerator = 1

1

1

1

1

>> numerator./B Warning: Divide by zero. ans = 5.0000e-001

1.0000e+000

Inf -1.0000e+000 -5.0000e-001

>> B=B+(B==0)*eps B = 2.0000e+000

1.0000e+000

2.2204e-016 -1.0000e+000 -2.0000e+000

>> numerator./B LogicalRelational_24 ans = 5.0000e-001

1.0000e+000

4.5036e+015 -1.0000e+000 -5.0000e-001

6RPH2WKHU:DUQLQJV« ‡ Using numeric values to represent logicals can have some VWUDQJHUHSHUFXVVLRQV« ‡ Never try to use NaN in a relational or logical expression because NaN has no value and therefore could be considered to have all values! aa = [ 10 16 16 16 17 >> aa > 15 ans = 0 1 >> aaa = ans >> aa(aaa) ans = 16 16 16 >> aaai = uint16(aaa) aaai = 0 1 1 >> aa(aaai) ??? Subscript indices integers or logicals.

9 11] 1

1

1

0

0

17

1

1

0

0

must either be real positive

>> nan==nan ans = 0 >> inf==inf ans = 1

Logical Functions ‡ Matlab includes a large number of logical functions (functions that return a true or false result) ‡ 0DQ\RIWKHPRVWXVHIXODUHZKDWDUHFDOOHG³LVD«´ functions ‡ see also functions: ± any( ) ± all( ) ± find( )

LogicalRelational_26

ages = [10 62 18 27] anyKids = any(ages <= 12) anySeniors = any(ages >= 65) anyKids =

allAdults = all(ages >= 18) noSeniors = all(ages <= 65) allAdults =

1

0

anySeniors =

noSeniors =

0

1

LogicalRelational_27

Summary ‡ Matlab can store and manipulate much more than simply numbers ‡ logical expressions on arrays ‡ GRQ¶WIRUJHW

Relational Operators: < less than <= less than or equal to > greater than >= greater than or equal to == equals to ~ = not equal to Logical Operators & AND | OR ~ NOT xor XOR

MATLAB interactive plotting

Topics ‡ Create various types of graphs ‡ Select variables to plot directly from a workspace browser ‡ Easily create and manipulate subplots in the figure ‡ Add annotations such as arrows, lines, and text ‡ Set properties on graphics objects

GraphicsInteractive_2

GraphicsInteractive_3

plotting your data ‡ Steps in plotting your data ‡ If only previewing or exploring data, steps 1 and 2 may be all you need. ‡ If creating presentation graphics, you may want to finetune your graph by positioning it on the page, setting line styles and colors, adding annotations, etc.

Prepare your data

Call elementary plotting function

Select line and marker Characteristics

Set axis limits, tick marks, and grid lines marks, and grid lines Annotate the graph with axis labels, legend and text

interactive plot editing ‡ Matlab s two ways to edit the plots you create: ± Using the mouse to select and edit objects interactively ± Using Matlab functions at the command line or in an M-file

‡ Interactive mode: ± perform point-and-click editing of graphs ± modify the appearance of a graphics object by double-clicking on the object and changing the values of its properties ± access the properties through a gui

GraphicsInteractive_4

GraphicsInteractive_6

Plotting Tools ‡ Start: ± use the plottools command. ± from the figure toolbar Show Plot Tools icon .

‡ 3 basic plotting tools ± Figure Palette, ± Plot Browser, ± Property Editor.

basic plotting tools ‡ Figure Palette: create and arrange subplot axes, view and plot workspace variables, and add annotations. ‡ Plot Browser: ± select and control the visibility of the axes or graphic objects plotted in the figure. ± add data to any selected axes by clicking the Add Data button.

‡ Property Editor: set common properties of the selected object.

GraphicsInteractive_7

GraphicsInteractive_8

Figure Palette / subplots ‡ New Subplots ² Add 2-D or 3-D axes to the figure. ± create a grid of either 2D or 3-D axes. ± click the grid icon next to the axes type. ± move the cursor, squares darken to indicate the layout of axes

GraphicsInteractive_9

Figure Palette / variables ‡ displays current workspace variables. ‡ double-clicking a variable opens that variable ‡ select a variable and rightclick to display the context menu, select a graphics function to plot the variable. ‡ If the desired plotting function is not available from the context menu, you can select More Plots to display the Plot Catalog tool. GraphicsInteractive_10

Figure Palette / annotations ‡ Add text, lines and arrows, rectangles, ellipses, and other annotation objects anywhere on the figure ‡ Anchor annotations to locations in data space ‡ Add a legend and colorbar ‡ Add axis labels and titles ‡ Edit the properties of graphics objects

basic annotations in Figure Palette

GraphicsInteractive_11

Figure Palette / annotations ‡ Annotation features are available from the Insert menu.

‡ Select Plot Edit Toolbar from the View menu to display the toolbar. GraphicsInteractive_12

Plot Browser ‡ provides a legend of all the graphs in the figure. It lists each axes and the objects (lines, surfaces, etc.) used to create the graph.

Plot Browser ‡ Controlling Object Visibility check box next to each item in the Plot Browser controls WKHREMHFW¶VYLVLELOLW\8QFKHFNWKHFROXPQV\RXGRQRW want to display. ‡ Deleting Objects delete any selected item in the Plot Browser by selecting Delete from the right-click context menu. ‡ Adding Data to Axes mechanism to add data to axes: ± Select a 2-D or 3-D axes from the New Subplots sub. ± After creating the axes, select it in the Plot Browser to enable the Add Data button at the bottom of the . ± Click the Add Data button to display the Add Data to Axes dialog.

GraphicsInteractive_13

Property Editor ‡ HQDEOHV\RXWRDFFHVVDVXEVHWRIWKHVHOHFWHGREMHFW¶V properties. ‡ when no object is selected, the Property Editor displays WKHILJXUH¶VSURSHUWLHV ‡ Ways to Display the Property Editor ± Double-click an object when plot edit mode is enabled. ± Select an object and right-click to display its context menu, then select Properties. ± Select Property Editor from the View menu.

GraphicsInteractive_14

Property Editor ‡ Property Editor enables you to change the most commonly used object ‡ select an object, the property editor changes accordingly ‡ to access all object properties, use the Property Inspector. GraphicsInteractive_15

Exporting using menus ‡ in a format that can be opened during another Matlab session ± Select Save from the figure window File menu or click the Save button on the toolbar. If this is the first time you are saving the file, the Save As dialog box appears. ± Make sure that the Save as type is Fig-File. ± Specify the name you want assigned to the figure file. ± Click OK.

GraphicsInteractive_16

Saving Your Work ‡ in a format that can be used by other applications ± Select the format from the list of formats in the Save as type drop-down menu. This selects the format of the exported file and adds the standard filename extension given to files of that type. ± Enter the name you want to give the file, less the extension. ± Click Save.

GraphicsInteractive_17

GraphicsInteractive_18

Generate code ‡ generate code to reproduce this graph by selecting Generate M-File from the Figure menu. Matlab creates a function that recreates the graph and opens the generated M-File in the editor. ‡ This feature is particularly useful for capturing property settings and other modifications made using the plot tools GUI. ‡ Data Arguments Generated functions do not store the data necessary to recreate the graph. You must supply the data arguments ‡ Limitations Attempting to generate code for graphs containing a large number of graphics objects (e.g., greater than 20 plotted lines) might be impractical.

MATLAB basic mathematical applictions

Topics ‡ basic data analysis ± mean, ± standard deviation, ....

‡ ‡ ‡ ‡ ‡

interpolation curve fitting polynomials solving linear equations function optimisation

basicMath_2

‡ Import Wizard for data import ± File-!,PSRUW'DWD«

‡ File input with load ± B = ORDGµGDWDLQW[W¶

‡ File output with save ± VDYHµdataout¶µ$¶µ-ascii¶

Data Analysis functions ‡ help datafun ‡ Search for: ± Data analysis

‡ ‡ ‡ ‡ ‡ ‡

Minimum of in a Data Set >> min(v) Maximum of in a Data Set >> max(v) Sum of a Data Set >> sum(v) Standard Deviation >> std(v) Mean of a Data Set >> mean(v) Sort a Data Set in ascending order>> sort(v)

‡ remark: MATLAB considers databasicMath_4 sets stored in column-oriented arrays

mean >> temp3city = 12 15 12 14 12 11 15 8 19 12 14 11 9 8 15 8 10 12 9 12 12 10 13 9 10 14 12 13 15 13 12

8 9 5 8 6 9 9 10 7 7 10 8 7 8 8 9 7 7 8 8 8 9 12 10 6 7 5 7 10 11 12

18 22 19 23 22 19 15 20 18 18 19 17 23 19 18 20 17 22 19 21 20 17 18 20 22 21 22 18 23 24 22

>> mean(temp3city, 1) ans = 11.9677 8.2258 >> mean(temp3city, 2) ans = 12.6667 15.3333 12.0000 15.0000 13.3333 13.0000 13.0000 12.6667 14.6667 12.3333 14.3333 12.0000 13.0000 11.6667 13.6667 12.3333 11.3333 13.6667 12.0000 13.6667 13.3333 12.0000 14.3333 13.0000 «

19.8710

‡ vectors: mean(X) is the mean value of the elements in X. ‡ matrices, mean(X) is a row vector containing the mean value of each column. ‡ mean(X,DIM) takes the mean along the dimension DIM of X.

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median ‡ median(x) same as mean(x), only returns the median value.

>> X = [0 1 2; 3 4 5] X = 0 3

1 4

2 5

>> median(X,1) ans = 1.5000 >> median(X,2) ans = 1 4

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2.5000

3.5000

std Standard deviation ‡ There are two common textbook definitions for the standard deviation s of a data vector X. ‡ Def_1 s

‡

n 1i 1

‡

‡

xi x

2

1 2

Def_2 s

‡

n

1

1 n

n i 1

xi x

2

1 2

s = std(X) X is a vector, returns the standard deviation using (1) above. X is a matrix, std(X) returns a row vector containing the standard deviation of the elements of each column of X. s = std(X,flag) flag = 0, is the same as std(X). flag = 1, std(X,1) returns the standard deviation using (2) s = std(X,flag,dim) computes the standard deviations along the dimension of X specified by scalar dim. Set flag to 0 to normalize Y by n-1; set flag to 1 to normalize by n

>> dat = 12 15 12 14 12 11 15 8 19 12 14 11 9 8 15 8 10 12 9 12 12 10 13 9 10 14 12 13 15 13 12

8 9 5 8 6 9 9 10 7 7 10 8 7 8 8 9 7 7 8 8 8 9 12 10 6 7 5 7 10 11 12

18 22 19 23 22 19 15 20 18 18 19 17 23 19 18 20 17 22 19 21 20 17 18 20 22 21 22 18 23 24 22

>> std(dat) ans = 2.5098 1.7646 2.2322 >> std(dat,1) ans = 2.4690 1.7360 2.1959 >> std(dat,0,2) ans = 5.0332 6.5064 7.0000 7.5498 8.0829 5.2915 3.4641 6.4291 6.6583 5.5076 4.5092 4.5826 8.7178 6.3509 5.1316 6.6583 5.1316 7.6376 4.3589 basicMath_7 ...

max (min) >> dat =

‡ max Largest component. ‡ vectors: MAX(X) is the largest element in X. ‡ matrices: MAX(X) is a row vector containing the maximum element from each column. ‡ [Y,I] = max(X) returns the indices of the maximum values in vector I. ‡ max(X,Y) returns an array the same size as X and Y with the largest elements taken from X or Y. Either one can be a scalar. ‡ [Y,I] = max(X,[],DIM) operates along the dimension DIM.

12 15 12 14 12 11 15 8 19 12 14 11 9 8 15 8 10 12 9 12 12 10 13 9 10 14 12 13 15 13 12

>> max(dat) 8 9 5 8 6 9 9 10 7 7 10 8 7 8 8 9 7 7 8 8 8 9 12 10 6 7 5 7 10 11 12

18 22 19 23 22 19 15 20 18 18 19 17 23 19 18 20 17 22 19 21 20 17 18 20 22 21 22 18 23 24 22

ans = 19

12

24

>> max(max(dat)) ans = 24 >> max(dat(:)) ans = 24 >> [Y, I] = max(dat)

Y = 19

12

24

9

23

30

I =

diff >> Y Y =

>> dat =

‡ Y = diff(X) calculates differences between adjacent elements of X. ‡ vector: returns a vector, 1 element shorter than X, of differences between adjacent elements: [X(2)-X(1) X(3)-X(2) ... X(n)-X(n-1)] ‡ matrix, then diff(X) returns a matrix of row differences: [X(2:m,:)-X(1:m-1,:)] ‡ Y = diff(X,n) applies diff recursively n times, diff(X,2) is same as diff(diff(X)). ‡ Y = diff(X,n,dim) is the nth difference function calculated along the dimension specified by scalar dim. If order n equals or exceeds the length of dimension dim, diff returns an empty array. basicMath_9

12 15 12 14 12 11 15 8 19 12 14 11 9 8 15 8 10 12 9 12 12 10 13 9 10 14 12 13 15 13 12

8 9 5 8 6 9 9 10 7 7 10 8 7 8 8 9 7 7 8 8 8 9 12 10 6 7 5 7 10 11 12

18 22 19 23 22 19 15 20 18 18 19 17 23 19 18 20 17 22 19 21 20 17 18 20 22 21 22 18 23 24 22

19 12 >> diff(Y) ans = -7 12 >> diff(Y,2) ans = 19 >> diff(dat) ans = 3 1 -3 -4 2 3 -2 -2 -1 3 4 0 -7 1 11 -3 -7 0 2 3 -3 -2 -2 -1 -1 1 7 0 -7 1 2 -2 2 0 -3 1 3 0 0 0 -2 1 3 3 -4 -2 1 -4 4 1 -2 -2 1 2 2 3 -2 1 -1 1

24

4 -3 4 -1 -3 -4 5 -2 0 1 -2 6 -4 -1 2 -3 5 -3 2 -1 -3 1 2 2 -1 1 -4 5 1 -2

filter filters a data sequence using a digital filter ‡ y = filter(b,a,X) filters the data in vector X with the filter described by numerator coefficient vector b and denominator coefficient vector a. If X is a matrix, filter operates on the columns of X. ‡ Algorithm: y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(nnb) - a(2)*y(n-1) - ... - a(na+1)*y(n-na) ‡ more detail in help filter

filter >> time_series = [12 17 10 22 15 11 18 27 14] time_series = 12 27

17 14

10

22

15

11

18

>> a = [.25 .50 .25] a = 0.2500

0.5000

0.2500

>> filter(a,1,time_series) ans = Columns 1 through 8 3.0000 17.2500

10.2500 15.7500

14.0000 13.7500

14.7500 18.5000

Column 9 21.5000

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more functions ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

corrcoef: correlation coeeficients cov: covariance matrix cumprod: cumulative product of elements cumsum: cumulative sum of elements histc(x, edges): histogram count and bin locations using bins marked by edges prod: product of elements sort: sorts in ascending or descending order sortrows: sort rows in ascending order sum: sum of elements demo_basic_DataAnalysis basicMath_12

Fitting/interpolation Tips: ‡ Check Cleve 0ROHU¶V website ± www.mathworks.com/moler

‡ Help ± Search in helpdesk data analysis ± Interpolation ± fitting

BasicDataAnalysis_13

Interpolation ‡ (VWLPDWHDYDULDEOH¶VYDOXHEHWZHHQNQRZQYDOXHV ± When you take data, how do you predict what other data points might be? ± Two techniques are : ‡ Linear Interpolation

‡ Cubic Spline Interpolation

‡ Extrapolation

Interpolation ‡ YI = interp1(X,Y,XI) interpolates to find YI, the values of the underlying function Y at the points in the vector or array XI ‡ Finding values between data points ±linear interpolation (default) ±cubic splines ±... yi = interp1 ( x, y, xi ) Yi = interp1 ( x, y, xi, 'spline' ) ±demo_interp1 ±demo_interp2 ±demo_interp3 BasicDataAnalysis_15

Fitting ‡ There is scatter in all collected data ‡ We can estimate what equation represents the GDWDE\³H\HEDOOLQJ´DJUDSK ‡ But there will be points that do not fall on the line we estimate ‡ method ± method of least squares ± minimize the residuals

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Linear regression ‡ Linear equation that is the best fir to a set of data points ‡ Minimize the sum of squared distances between the line and the data points ‡ demo_linreg1.m

Polynomial Fit ‡ polyfit finds the coefficients of a polynomial representing the data ‡ JHQHUDWHVD³EHVWILW´SRO\QRPLDOLQWKHOHDVW squares sense) of a specified order for a given set of data. ‡ used to generate the n + 1 coefficients aj of the nth-degree polynomial used for fitting the data. pn (x) an xn an 1xn 1 K an x a0

‡ polyval: uses those coefficients to find new values of y, that correspond to the known values of x basicMath_18

polyfit >> x = [0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1]; >> y = [-.447 1.978 3.28 6.16 7.08 7.34 7.66 9.56 9.48 9.30 11.2]; >> n = 2; >> p = polyfit(x,y,n); >> p = -9.8108 20.1293 -0.0317 xi = linspace(0,1,100); yi = polyval(p,xi); pp = polyfit(x,y,10); y10 = polyval(pp,xi); plot(x,y,'o',xi,yi,'-',xi,y10) % plot data xlabel('x'), ylabel('y=f(x)') title('2nd and 10th Order Curve Fitting') demo_polyfit basicMath_19

built-in tool

basicMath_20

polynomials ‡

Polynomials are described by using a row vector of the coefficients of the polynomial beginning with the highest power of x and inserting zeros for ³PLVVLQJ´WHUPV f=[9 ±5 3 7]; g=[6 ±1 2]; ‡

add and subtract polynomial functions in MATLAB. To do this we must ³]HUR´SDGWKHSRO\QRPLDOVVRWKDWWKHLUURZYHFWRUUHSUHVHQWDWLRQVDUHWKH same length: h=f+[0 g]; ‡ multiply and divide polynomials by using the conv and deconv functions y=conv(f,g) [q r]=deconv(y,f) ‡ evaluate a polynomial at any value of x: p=[1 3 -4]; x=[-5:.1:5]; px=polyval(p,x); basicMath_21

polynomials ‡ roots finds polynomial roots. roots( [ 1 6 11 6 ] ) ans = -3.0000 -2.0000 -1.0000 ‡ polyder(P) returns the derivative of the polynomial whose coefficients are the elements of vector P. >> polyder( [ 1 6 11 6 ] ) ans = [ 3 12 11 ]

Solution?

2x1 3x2 8 5x1 4x2 13 A = [2 3; 5 4] b = [8;13] x = A\b

LinearEquations_23

Solution?

2 x1 3x2 4 4 x1 6 x2 7 A = [2 3; 4 6] b = [4;7] x = A\b

LinearEquations_24

Simultaneous Linear Equations ‡ Basic form: a11x1 + a12x2 + a13x3 «D1nxn = b1 a21x1 + a22x2 + a23x3 «D2nxn = b2 ««« an1xn + an2x2 + an3x2 «Dnnxn = bn

a11 a12 a21 a22

a1n x1 a2n x2

b1 b2

‡ Where:

an1 an 2

ann xn

bn

± aij are known coefficients ± xi are unknowns ± bi are known right hand side values

or Ax b

LinearEquations_25

Solution of equations 2x1 +4x2-2x3=4 4 x1+9x2-2x3=6 -2 x1-3x2+7x3=10 What is the solution? demo_lineq3

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‡ The coefficient matrix A need not be square. If A is m*n, there are three cases. ± m=n ‡ Square system. ‡ Seek an exact solution.

± m>n ‡ Over determined system. ‡ Find a least squares solution.

± m
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Linear Equations ‡ Set of linear equations Ax = b can be solved using the Inverse operation on A x=inv(A)*b

‡ Advice: ± Do not use this method ± Inefficient

basicMath_28

backslash \ ‡ MATLAB has a bunch of methods available to solve a system of linear equations ‡ If you desire the solution of Ax = b, then the simplest method using Matlab to find x is to set x = A\b A = [ 1 5 6; 7 9 6; 2 3 4] b = [29; 43; 20] x=A\b

‡ Use backslash operator \ (LU decumposition + pivoting) ‡ If A is an n by m matrix and b is an p by q matrix then A\b is defined (and is calculated by Matlab) if m=p. ‡ For non-square and singular systems, the operation A\b gives the solution in the least squares sense. No error message ‡ 7KH³\ ´EDFN-slash operator uses a combination of numerical methods including LU decomposition.

Steps in A\b Stop if successful ‡ If A is upper or lower triangular, solve by back/forward substitution ‡ If A is permutation of triangular matrix, solve by permuted back substitution (useful for [L,U]=lu(A) since L is permuted) ‡ If A is symmetric/hermitian ± Check if all diagonal elements are positive ± Try Cholesky, if successful solve by back substitutions

‡ If A is Hessenberg (upper triangular plus one subdiagonal), reduce to upper triangular then solve by back substitution ‡ If A is square, factorize PA = LU and solve by back substitutions ‡ If A is not square, run Householder QR, solve least squares problem

Basic optimization ‡ optimization means here: determine where a function y = f(x) takes on either specfic or extreme values ‡ Zero finding ‡ Minimization in 1 dimension

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Zero finding ‡ fzero: Find zero of a function of one variable find x satisfying f(x) = 0 ‡ $QLPSOHPHQWDWLRQRI7'HNNHU¶VDOJRULWKP ‡ Combination of ± Secant method / quadratic fit extension of Secant method ± Bisection

‡ While not universally effective, combining techniques can improve the reliability and speed of convergence ‡ fzero uses bisection to avoid large steps, other methods for fast convergence ‡ syntax: x = fzero(fun,x0) ± fun supplied function specifying f(x) ± x0 is an initial guess of an x

Challenges ‡ Finding good initial values ± For 1D functions, graphing can be an aid ± Graphing not always possible for some multi-D functions, or even time-consuming 1D functions

‡ Analytic methods, if feasible, are always a better choice than numeric algorithms! ‡ demo_fzero_1

Minimization in 1 dimension ‡ X = fminbnd(FUN,x1,x2) attempts to find a local minimizer X of the function FUN in the interval x1 < X < x2. FUN accepts scalar input X and returns a scalar function value F evaluated at X. ‡ demo_fminbnd ‡ demo_fminbnd_2

Maximum? ‡ Find maximum of:

y

1 1 5 2 3 ( x 0.4) 0.01 ( x 0.7) 0.04

‡ demo_fminbnd_3

Minimization in n dimensions ‡ X = fminsearch(FUN,X0) starts at X0 and attempts to find a local minimizer X of the function FUN. FUN accepts input X and returns a scalar function value F evaluated at X. X0 can be a scalar, vector or matrix. ‡ Multidimensional unconstrained nonlinear minimization (Nelder-Mead).