NatSciLab - Numerical Software Introduction to MATLAB

Transcription

1 Outline NatSciLab - Numerical Software Introduction to MATLAB Onur Oktay Jacobs University Bremen Spring 2010

2 Outline 1 MATLAB Desktop Environment 2 The Command line A quick start Indexing 3 Operators Arithmetic (Array and Matrix) Operators Logical and Relational Operators 4 MATLAB data types MATLAB Number format: Floating Point

3 Outline 1 MATLAB Desktop Environment 2 The Command line A quick start Indexing 3 Operators Arithmetic (Array and Matrix) Operators Logical and Relational Operators 4 MATLAB data types MATLAB Number format: Floating Point

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5 Command Window is where you type the commands and view the outputs. Command History records the commands you type at the command line. - Double-click on a command to repeat it - Right-click on a command to cut/copy/delete it Workspace shows all of the variables that are currently defined in your Matlab session. - Double-click on a variable to view it in the array editor. - Right-click on a variable to rename/delete/etc. it Matlab Editor is a specialized text editor for m-files.

6 Outline 1 MATLAB Desktop Environment 2 The Command line A quick start Indexing 3 Operators Arithmetic (Array and Matrix) Operators Logical and Relational Operators 4 MATLAB data types MATLAB Number format: Floating Point

7 When you start MATLAB, you see the prompt >> This is the command line. As we go through the slides, please copy the commands after each prompt to your MATLAB command line. We start with >> diary on MATLAB just formed a diary.txt under your current directory. Everything on the command window is copied into diary.txt until you type diary off.

8 + / ˆ We can do simple arithmetic operations ( +,, *, /,ˆ) on the command line such as >> ( ˆ2 4*7 ) / 12 ans = We can also do this calculation by assigning variable names. >> x = 2+3.5ˆ2 x = >> y = 4*7 y = 28 >> z = (x-y)/12 z =

9 ; If we do not want to see the intermediate results, we can suppress the output by putting a semicolon (;) at the end of the line. For example, with the semicolons, the sequence of commands we just entered and their outputs look like >> x = 2+3.5ˆ2; >> y = 4*7 ; >> z = (x-y)/12 ; >> z >> z =

10 If we enter an expression incorrectly, MATLAB returns an error message. For example, in the following, we left out the * in 3*x. >> x = 4; >> 3x??? 3 Missing operator, comma, or semicolon. >> Another example: >> 2*(x+y??? 2*(x+y A closing right parenthesis is missing. Check for a missing ) or a missing operator. >>

11 MATLAB recognizes all variables as arrays. Numbers are 1 1 arrays. Vectors are 1 n or n 1 arrays. Matrices are m n arrays. 3D, 4D,..., n-dimensional d 1 d 2 d n arrays. >> ThreeD = rand(5,4,3); Non-numeric arrays: logical, struct, cell arrays (more is coming soon)

12 Matrices can be formed by typing in the elements one at a time. - Type the entries in brackets [ ]. - Use semicolon ; to separate the rows. >> u = [ ]; >> u = >> A = [ ; ; ; ] A =

13 Usage Regularly spaced 1 n vectors are formed by the colon operator. vector = start:increment:end % increment = 1 by default, if not provided. Usage >> u2 = 1:3:13 % same as u2 = [ ] >> v = 1: -0.2:0; % same as v = [ ] >> v = 1:5; % same as v=[ ] Regularly spaced 1 n vectors are formed also by linspace. v = linspace(a,b,n) % Generates a 1 N vector v of equally spaced points % between and including a and b >> v2 = linspace(0,1,11);

14 zeros(m,n) produces a m n matrix of 0 s. zeros(n) produces a n n square matrix of 0 s. ones(m,n) produces a m n matrix of 1 s. ones(n) produces a n n square matrix of 1 s. eye(m,n) returns a m n matrix with 1 s on the diagonal and 0 s elsewhere. eye(n) produces a n n identity matrix. diag(v,k) when v is a vector of n components, returns a (n + k ) (n + k ) square matrix, with the elements of v on the kth diagonal. k = 0 is the main diagonal, k > 0 above the main diagonal, and k < 0 below the main diagonal. diag(a,k) for matrix A, returns a column vector v formed from the elements of the kth diagonal of A. >> m=4; n=7; >> D=diag(v2) >> v3=diag(d) >> ones(m,n) >> eye(m,n)

15 Indexing >> u(2) returns the 2nd entry of the vector u >> u(2,2) gives an error message >> A(1,3) returns the 1st row 3rd column entry of the matrix A. >> A(1,:) returns the 1st row of A >> A(:,2) returns the 2nd column of A >> A(:,[1 3]) returns the submatrix formed by the 1st and 3rd columns. >> A( [1 3 4], 2:5 ) forms the submatrix of A with the entries >> a = A(:); stacks up the columns of the matrix A

16 Indexing If u is a vector of length n, and M is a matrix with integer entries between 1 n, then u(m(1, 1)) u(m(1, 2))... u(m(1, n)) u(m(2, 1)) u(m(2, 2))... u(m(2, n)) u(m) = u(m(m, 1)) u(m(m, 2))... u(m(m, n)) >> u=1:3:13; w = u([1 3 4]) % vector with entries u(1),u(3),u(4) >> ind = [2 2; 4 1; 5 3]; u(ind) ans =

17 Linear Indexing Elements of a matrix can be called with a single index A(k). If A is m n, then A(k) = A(i,j) where k = (j 1) m + i A(k) = a(k), where a = A(:) 1 m + 1 2m (n 1)m m + 2 2m (n 1)m m + 3 2m (n 1)m m 2m 3m... nm

18 find find locates all nonzero elements of a numerical array or logical expression, and returns their linear indices. >> B = 1-2*rand(3,4); indb = find(b) >> C = magic(4); C C = >> indc = find(c>12) % indices k for which C(k)> 12 indc = We can do the same task by logical indexing (next slide)

19 Logical indexing Matlab provides you with all elements of a matrix that meet a specified condition. >> v = [ ] ; >> indv-log = (v > 2) % forms a logical variable indv-log = % 0 = false, 1 = true >>v(indv-log) ans = >> indv = find(v > 2) ans = >>v(indv) ans = indv gives indices k for which v(k) > 2 indv-log(k)=1 if v(k) meets the condition, 0 otherwise. Note: v(indv) and v(indv-log) are the same.

20 Logical indexing Example: Find all entries of A2 that are < and set them to 0. >> A2 = [0.3, 4, 7; 2, 1e-5, 5; 3e-6, 4.2 4e-9] ; >> inda2-log = (A2 < 0.001) inda2-log = >> A2(indA2-log) = 0 A2 = We can do the same quickly in 1-line as below: >> A2(A2< 0.001) = 0; >> A2(find(A2< 0.001)) = 0;

21 Outline 1 MATLAB Desktop Environment 2 The Command line A quick start Indexing 3 Operators Arithmetic (Array and Matrix) Operators Logical and Relational Operators 4 MATLAB data types MATLAB Number format: Floating Point

22 Array/Matrix operations Operation Description + Addition Subtraction. Entrywise multiplication./ Entrywise division Matrix multiplication / Matrix right division \ Matrix left division. Entrywise power Matrix Power. Transpose Complex conjugate transpose To make an operation array smart, put a dot before it. Note: Operations, which are the same for matrices & arrays, do not need a dot (+ )

23 Array/Matrix operations >> m=5; n=7; r=4; b=5; >> A = (rand(m,n)>0.5); % m n matrix >> B = [ones(m-2,3), eye(m-2,n-3); zeros(2,3), ones(2,n-3)]; >> C = randi(n,r); % n r matrix >> AplusB = A+B; AminusB = A-B; % must have size(a) = size(b) >> AC = A C; % must have size(a,2) = size(c,1) >> A B % error message follows >> AtimeB = A. B; AoverB = A./B; % must have size(a) = size(b) >> A. C % error message follows >> ba = b A; bplusa = b+a; % mult./add b to every element of A >> S = diag(1:m) + diag(ones(m-1,1),1); % m m square matrix >> S2 = Sˆ2; % same as S S >> S3 = S.ˆ2; % same as S. S

24 Matrix division and inversion Left matrix division mldivide: A is m n, B is m r. X = A\B is the least squares solution to the matrix equation A X = B. Right matrix division mrdivide: A is n m, B is r m. X = A/B is the least squares solution to the matrix equation X A = B. inv(s) calculates the inverse of a square matrix S. Note: A/B = ( B \A ) If A is a nonsingular square matrix, then X=A\B is equal to inv(a) B.

25 Matrix division and inversion >> A = [1 0 0; 0 1 0; 1 1 1]; >> b = [ 1; 0; -1]; >> x = A\b x = >> x = inv(a)*b >> A = [1 1; 1 3; 0 1]; >> b = [1; 0; 0]; x = A\b; >> lserror = A x - b >> b = [5; 11; 3]; x = A\b; >> lserror = A x - b

26 Logical and relational operators Operator Description < Less than <= Less than or equal to > Greater than >= Greater than or equal to == Equal to = Not equal to & Logical AND Logical OR Logical NOT

27 Relational operators < <= > >= == = - Always work with numerical arrays, - Always operate entrywise. - Output is always of the type logical. >> A = [5 4 1; 9 0 5; ]; >> B = [8 7 1; 9 2 5; ]; >> A == B ans = >> q = (A< 5) q =

28 Logical operators & Always operate entrywise. For each entry, both nonzero one of them nonzero both zero A & B A B A N/A 1 0 >> A = [1, 0, 0, 1, 0, 1, 4, 5, -2]; >> B = [1, 1, 0, 0, 1, 1, 6, 0, 9.253]; >> AandB = A&B AandB = >> AorB = A B AorB = >> Anegation = A Anegation =

29 Logical operators Other functions that return logical values - isempty(var): 1 if var=[], 0 else - isequal(a,b): 1 if the numer. arrays A and B are equal, 0 else. - isreal(var): 1 if var is real, 0 else - isinf(var): 1 if var=inf, 0 else - isinf(var): 1 if var=inf, 0 else See also isnumeric, islogical, ischar, iscell, isstruct, isfloat, isinteger, isa. >> help isa

30 Outline 1 MATLAB Desktop Environment 2 The Command line A quick start Indexing 3 Operators Arithmetic (Array and Matrix) Operators Logical and Relational Operators 4 MATLAB data types MATLAB Number format: Floating Point

31 MATLAB classes (data types) Class Name double, single int8, uint8, int16, uint16, int32, uint32, int64, uint64 char logical function-handle struct cell Intended Use Default numeric type. Range = e-308 to e+308 Integers. Signed/unsigned. 8,16,32,64 bits. Characters and Strings Boolean type. Can have 2 values: true or false. Pointer to a function. Structures. Fields store arrays of varying classes and sizes. Cell Arrays. Cells store arrays of varying classes and sizes.

32 MATLAB Number format MATLAB does numerical calculations in double precision, which is 15 digits. Normally only 5 digits are displayed. If we want to see all 15 digits, >> format long >> z = 26/3 z = To return to the short format, >> format short

33 MATLAB Number format A real number x can be viewed as x = s m 10 r where 0 m< 1 and s= ±. In the double-precision format all numbers are stored in 64-bit words. A 64-bit word is a sequence (b 1 b 2... b 64 ) of 0 s and 1 s. The first bit b 1 gives the sign s (0 = +, 1 = ), The next 11 bits encode the exponent r between = 1023 and 2 10 = 1024 so that r = b 2 + b b b b The remaining 52 bits are used to describe the fraction or mantissa m so that m = b 13 + b b b

34 MATLAB Number format Consequently, there are finitely many machine numbers. There are little gaps between any two neighboring machine numbers. The bigger the numbers become in absolute value, the larger become the gaps. realmax is the biggest positive number in MATLAB, realmin is the smallest positive number in MATLAB, eps is the distance from 1 to the next larger double precision number, that is eps= 2 52 x>realmax x = Inf x<-realmax x = Inf indeterminate foms such as 0/0 or Inf/Inf are assigned to NaN

35 All for today >> diary off