Rutgers University School of Engineering Spring 2015 14

Rutgers University School of Engineering Spring 2015 14: 440: 127 - Introduction to Computers for Engineers Sophocles J. Orfanidis ECE Department orfanidi@ece. rutgers. edu week 13

Weekly Topics Week 1 - Basics – variables, arrays, matrices, plotting (ch. 2 & 3) Week 2 - Basics – operators, functions, strings, cells (ch. 2, 3, 11) Week 3 - Matrices (ch. 4) Week 4 - Plotting – 2 D and 3 D plots (ch. 5) Week 5 - User-defined functions (ch. 6) Week 6 - Input-output formatting – fprintf, fscanf (ch. 7) – Exam 1 Week 7 - Relational & logical operators, if, switch statements (ch. 8) Week 8 - For-loops, while-loops (ch. 9) Week 9 - Review for weeks 5 -8 Week 10 - Matrix algebra – solving linear equations (ch. 10) – Exam 2 Week 11 - Numerical methods I (ch. 13) Week 12 - Numerical methods II (ch. 13) Week 13 - Strings, structures, cell arrays, symbolic math (ch. 11, 12) Week 14 – Exam 3 References: H. Moore, MATLAB for Engineers, 4/e, Prentice Hall, 2014 G. Recktenwald, Numerical Methods with MATLAB, Prentice Hall, 2000 A. Gilat, MATLAB, An Introduction with Applications, 4/e, Wiley, 2011

Strings, Structures, Cells, Symbolic Math • • characters and strings concatenating strings using num 2 str comparing strings with strcmp • structure arrays • converting structures to cells • cell arrays • cell vs. content indexing • varargin, varargout • multi-dimensional arrays • symbolic math

MATLAB Data Classes Character Logical Numeric Integer signed unsigned user-defined classes Cell Structure Floating Point single precision More Classes Java classes Symbolic function handles double precision Cell and Structure arrays can store different types of data in the same array

Characters and Strings >> c = 'A' c = A Strings are arrays of characters. Characters are represented internally by standardized numbers, referred to as ASCII (American Standard Code for Information Interchange) codes. see Wikipedia link: ASCII table >> x = double(c) x = 65 % ASCII code for 'A' >> char(x) ans = A >> class(c) ans = char() creates a character string >> doc char >> doc class

>> s = 'ABC DEFG' s = ABC DEFG >> x = double(s) x = 65 66 67 32 68 69 70 71 >> char(x) ans = ABC DEFG >> size(s) ans = 1 8 >> class(s) ans = char ASCII codes convert ASCII codes to characters s is a row vector of 8 characters >> s(2), s(3: 5) ans = B ans = C D

Concatenating Strings s = ['Albert', 'Einstein'] s = Albert. Einstein >> s = ['Albert', ' Einstein'] s = Albert Einstein preserve leading and trailing spaces >> s = ['Albert ', 'Einstein'] s = >> doc Albert Einstein >> doc >> size(s) >> doc ans = >> doc 1 15 strcat strvcat num 2 str strcmp findstr

Concatenating Strings s = strcat('Albert ', 'Einstein') s = strcat strips trailing spaces Albert. Einstein but not leading spaces >> s = strcat('Albert', ' Einstein') s = Albert Einstein use repmat to make up long format strings for use with fprintf >> fmt = strcat(repmat('%8. 3 f ', 1, 6), 'n') fmt = %8. 3 fn

Concatenating Vertically s = ['Apple'; 'IBM'; 'Microsoft']; ? ? ? Error using ==> vertcat CAT arguments dimensions are not consistent. s = ['Apple '; 'IBM s = Apple IBM Microsoft padded with spaces to the length of the longest string >> size(s) ans = 3 9 '; 'Microsoft']

Concatenating Vertically s = strvcat('Apple', 'IBM', 'Microsoft'); s = char('Apple', 'IBM', 'Microsoft'); s = Apple IBM Microsoft >> size(s) ans = 3 9 strvcat, char both pad spaces as necessary Recommendation: use char to concatenate vertically, and [] to concatenate horizontally

a = [143. 87, -0. 0000325, -7545]'; num 2 str >> s = num 2 str(a) s = num 2 str(A) 143. 87 s = num 2 str(A, precision) -3. 25 e-005 s = num 2 str(A, format) -7545 >> s = num 2 str(a, 4) s = 143. 9 max number of digits -3. 25 e-005 -7545 >> s = num 2 str(a, '%12. 6 f') s = 143. 870000 format spec -0. 000032 -7545. 000000

a = [143. 87, -0. 0000325, -7545]'; >> s = num 2 str(a, '%10. 5 E') s = 1. 43870 E+002 -3. 25000 E-005 -7. 54500 E+003 b = char('A', 'BB', 'CCC'); >> disp([b, repmat(' A 1. 43870 E+002 BB -3. 25000 E-005 CCC -7. 54500 E+003 ', 3, 1), s]) num 2 str

t = linspace(0, 5, 101); w = [2, 5, 8, 10]; Y = sin(w'*t); % 4 x 101 matrix s = char('b-', 'r-', 'm-', 'g-'); for i=1: 4, figure; plot(t, Y(i, : ), s(i, : )); title(['sin(', num 2 str(w(i)), 't)']); print('-depsc', ['file', num 2 str(i), '. eps']); end Labeling and saving multiple plots with num 2 str saved as file 1. eps, file 2. eps file 3. eps, file 4. eps

Comparing Strings are arrays of characters, so the condition s 1==s 2 requires both s 1 and s 2 to have the same length >> s 1 = 'short'; s 2 = 'shore'; >> s 1==s 1 ans = 1 1 1 >> s 1==s 2 ans = 1 1 0 >> s 1 = 'short'; s 2 = 'long'; >> s 1==s 2 ? ? ? Error using ==> eq Matrix dimensions must agree.

Comparing Strings Use strcmp to compare strings of unequal length, and get a binary decision >> s 1 = 'short'; s 2 = 'shore'; >> strcmp(s 1, s 1) ans = >> doc strcmp 1 >> doc strcmpi >> strcmp(s 1, s 2) ans = 0 >> s 1 = 'short'; s 2 = 'long'; >> strcmp(s 1, s 2) ans = 0 case-insensitive

Useful String Functions sprintf sscanf deblank strcmpi strmatch upper lower blanks strjust strtrim strrep findstr - write formatted string - read formatted string - remove trailing blanks - compare strings - find possible matches - convert to upper case - convert to lower case - string of blanks - left/right/center justify string - remove leading/trailing spaces - replace strings - find one string within another

Structures name. field Structures have named ‘fields’ that can store all kinds of data: vectors, matrices, strings, cell arrays, other structures student. name = 'Apple, A. '; student. id = 12345; student. exams = [85, 87, 90]; student. grades = {'B+', 'A'}; >> student = name: id: exams: grades: 'Apple, A. ' 12345 [85 87 90] {'B+' 'A'} % % string number vector cell >> class(student) ans = structures can also be created with struct()

Structure Arrays structure array index add two more students with partially defined fields, rest of fields are still empty student(2). name = 'Twitter, T. '; student(3). id = 345678; >> student = 1 x 3 struct array with fields: name id exams grades

Structure Arrays >> student(1) ans = name: 'Apple, A. ' id: 12345 exams: [85 87 90] grades: {'B+' 'A'} >> student(2) ans = name: 'Twitter, T. ' id: [] exams: [] grades: [] >> student(3) ans = name: [] id: 345678 exams: [] grades: []

Structure Arrays the missing field values can be defined later and don’t have to be of the same type or length as those of the other entries >> student(3). exams = [70 80]; >> student(3) ans = name: [] id: 345678 exams: [70 80] grades: []

Accessing Structure Elements >> student(1) ans = name: 'Apple, A. ' id: 12345 exams: [85 87 90] grades: {'B+' 'A'} >> student(1). name(5) >> student(1). exams(2) >> student(1). grades(3) >> student(1). grades{3} % % % % ans = e ans = 87 ans = 'A' ans = A cell vs. content indexing

s. a = [1 2; 3 4]; s. b = {'a', 'bb'; 'ccc', 'dddd'}; >> s s = a: [2 x 2 double] b: {2 x 2 cell} >> s. a ans = 1 3 >> s. a(2, 2) ans = 4 2 4 >> s. b ans = 'a' 'ccc' 'bb' 'dddd' >> s. b(2, 1), s. b{2, 1} ans = cell vs. 'ccc' content ans = indexing ccc

Nested Structures student. name = 'Apple, A. '; student. id = 12345; student. work. exams = [85, 87, 90]; student. work. grades = {'B+', 'A'}; >> student = name: 'Apple, A. ' id: 12345 work: [1 x 1 struct] >> student. work % sub-structure ans = exams: [85 87 90] grades: {'B+' 'A'}

Stucture Functions struct fieldnames isstruct isfield rmfield struct 2 cell - create structure get structure field names test if a structure test if a field remove a structure field convert structure to cell array >> C = struct 2 cell(student) C = 'Apple, A. ' [ 12345] [1 x 1 struct] >> C{3} content indexing ans = exams: [85 87 90] grades: {'B+' 'A'}

Like structures, cell arrays are containers of all kinds of data: vectors, matrices, strings, structures, other cell arrays, functions. Cell Arrays A cell is created by putting different types of objects in curly brackets { }, e. g. , c = {A, B, C, D}; c = {A; B; C; D}; c = {A, B; C, D}; % 1 x 4 cell % 4 x 1 cell % 2 x 2 cell where A, B, C, D are arbitrary objects c{i, j} accesses the data in i, j cell c(i, j) is the cell object in the i, j position cell vs. content indexing

A B C D = = {'Apple'; 'IBM'; 'Microsoft'}; [1 2; 3 4]; @(x) x. ^2 + 1; [10 20 30 40 50]; c = {A, B; C, D} c = {3 x 1 cell} @(x)x. ^2+1 % % cells matrix function row % define 2 x 2 cell array [2 x 2 double] [1 x 5 double] >> cellplot(c);

A = {'Apple'; 'IBM'; 'Microsoft'} A = 'Apple' 'IBM' 'Microsoft' comparing cell arrays of strings vs. strings >> size(A), class(A) ans = 3 1 ans = cell S = char('Apple', 'IBM', 'Microsoft') S = Apple IBM Microsoft >> size(S), class(S) ans = 3 9 ans = char

>> A(2), class(A(2)) ans = 'IBM' ans = cell vs. content >> A{2}, class(A{2}) indexing ans = IBM ans = char >> A' ans = 'Apple' 'IBM' 'Microsoft' >> S Apple IBM Microsoft >> S' AIM p. Bi p. Mc l r e o s o f t

A B C D = = {'Apple'; 'IBM'; 'Microsoft'}; [1 2; 3 4]; @(x) x. ^2 + 1; [10 20 30 40 50]; c = {A, B; C, D} c = {3 x 1 cell} @(x)x. ^2+1 % % cells matrix function row % define 2 x 2 cell array [2 x 2 double] [1 x 5 double] >> cellplot(c);

content indexing with { } >> celldisp(c) c{1, 1}{1} = Apple c{1, 1}{2} = IBM c{1, 1}{3} = Microsoft >> c{1, 1} ans = 'Apple' 'IBM' 'Microsoft' c{2, 1} = @(x)x. ^2+1 c{1, 2} = 1 3 2 4 c{2, 2} = 10 20 >> c{2, 1} ans = @(x)x. ^2+1 30 40 50

>> c{1, 1}{3} ans = Microsoft >> c{1, 1}{3}(6) ans = s content indexing with { } >> c{1, 2}(2, : ) ans = 3 4 >> x = [1 2 3]; >> c{2, 1}(x) ans = 2 5 10 >> c{1, 2}(1, 2) ans = 2 >> fplot(c{2, 1}, [0, 3]); >> c{2, 2}(3) ans = 30

cell indexing ( ) content indexing { } >> c(2, 2) cell ans = [1 x 5 double] >> class(c(2, 2)) ans = cell >> c{2, 2} ans = 10 20 cell contents 30 >> class(c{2, 2}) ans = double 40 50

>> c{1, 1}(2) ans = 'IBM' cell >> class(c{1, 1}(2)) ans = cell >> c{1, 1}{2} ans = IBM cell contents >> class(c{1, 1}{2}) ans = char cell indexing ( ) content indexing { }

>> d = c; >> % d(1, 3) = {[4 5 6]'}; >> d{1, 3} = [4, 5, 6]' % define as cell % define content d = {3 x 1 cell} @(x)x. ^2+1 [3 x 1 double] [] [2 x 2 double] [1 x 5 double] >> d(2, 3) ans = {[]} >> d{2, 3} ans = [] >> cellplot(d);

>> e = reshape(d, 3, 2) >> cellplot(e) >> f = repmat(c, 1, 2) >> cellplot(f) try also >> f = d';

>> c{1, 1}{2} = 'Google'; changing cell >> c{1, 2} = 10*c{1, 2}; array contents >> c{2, 2}(3) = 300; >> celldisp(c) c{1, 1}{1} = could have used: Apple c{1, 1}(2) = {'Google'}; c{1, 1}{2} = Google c{1, 1}{3} = why not ? Microsoft c(1, 2) = 10*c(1, 2); c{2, 1} = @(x)x. ^2+1 why not ? c{1, 2} = c{2, 2}{3} = 300; 10 20 30 40 c{2, 2} = 10 20 300 40 50

num 2 cell 2 mat converts numerical cell arrays to numbers num 2 cell converts numerical array to cell array >> A = [1 2; 3 4]; >> C = num 2 cell(A) C = [1] [3] [2] [4] >> B = cell 2 mat(C) B = 1 3 2 4

>> A = [1, 2; 3, 4]; % 2 x 2 matrix >> S = {'a', 'bb'; 'ccc', 'dddd'}; num 2 cell 2 mat % 2 x 2 cell S = 'a' 'ccc' 'bb' 'dddd' >> [A, S] % can’t mix numbers and cells ? ? ? Error using ==> horzcat CAT arguments dimensions are not consistent. >> [num 2 cell(A), S] ans = [1] [3] [2] [4] % construct 2 x 4 cell 'a' 'ccc' 'bb' 'dddd'

using num 2 cell with fprintf S = {'January', 'April', 'July', 'October'}; T = [30, 60, 80, 70]; C = ? ? ? CAT are [S; T] % can’t mix numbers and cells Error using ==> vertcat arguments dimensions not consistent. C = [S; num 2 cell(T)] % make all into cells C = 'January' [ 30] fprintf(' January April July October 'April' [ 60] %-7 s 30. 00 60. 00 80. 00 70. 00 'July' [ 80] 'October' [ 70] %5. 2 fn', C{: }); read content column-wise print it row-wise >> class(C) ans = cell >> C(: ) ans = 'January' [30] 'April' [60] 'July' [80] 'October' [70]

M = {'January'; 'April'; 'July'; 'October'}; T 1 = [30; 60; 80; 70]; column vectors T 2 = [-10; 40; 60; 50]; transposed C = [M, num 2 cell([T 1, T 2])]' % C = % 'January' % [ 30] % [ -10] num 2 cell fprintf >> C(: ) 'April' [ 60] [ 40] 'July' [ 80] [ 60] 'October' [ 70] [ 50] fprintf(' T 1 T 2n') fprintf('-------------n') fprintf('%-7 s %6. 2 fn', C{: }) % fprintf('%-7 s %6. 2 fn', C{: , : }) T 1 T 2 ------------January 30. 00 -10. 00 April 60. 00 40. 00 July 80. 00 60. 00 October 70. 00 50. 00 content indexing ans = 'January' [ 30] [-10] 'April' [ 60] [ 40] 'July' [ 80] [ 60] 'October' [ 70] [ 50]

varargin varargout varargin, varargout are cell arrays that allow the passing a variable number of function inputs & outputs % [x, y, vx, vy] = trajectory(t, v 0, th 0, g) function [varargout] = trajectory(t, v 0, varargin) Nin = nargin-2; % number of varargin inputs if Nin==0, th 0=90; h 0=0; g=9. 81; end if Nin==1, th 0=varargin{1}; h 0=0; g=9. 81; end if Nin==2, th 0=varargin{1}; . . . h 0=varargin{2}; g=9. 81; end if Nin==3, th 0=varargin{1}; . . . h 0=varargin{2}; g=varargin{3}; end continues

th 0 = th 0 * pi/180; % convert to radians x = v 0 * cos(th 0) * t; y = h 0 + v 0 * sin(th 0) * t - 1/2 * g * t. ^2; vx = v 0 * cos(th 0); vy = v 0 * sin(th 0) - g * t; if nargout==1; varargout{1} if nargout==2; varargout{1} varargout{2} if nargout==3; varargout{1} varargout{2} varargout{3} if nargout==4; varargout{1} varargout{2} varargout{3} varargout{4} = = = = = x; end x; . . . y; end x; y; . . . vx; end x; . . . y; . . . vx; . . . vy; end

Multidimensional Arrays A three-dimensional array is a collection of two-dimensional matrices of the same size, and are characterized by triple indexing, e. g. , A(i, j, p) is the (i, j) matrix element of the p-th matrix. Higher-dimensional arrays can also be defined, e. g. , a 4 D array is a collection of 3 D arrays of the same size. applications in video processing

>> >> a = [1 2; 3 4]; A(: , 1) = a; A(: , 2) = 10*a; A(: , 3) = 100*a; >> A A(: , 1) = 1 2 3 4 A(: , 2) = 10 20 30 40 A(: , 3) = 100 200 300 400 pages sum, min, max can operate along the i, j, p dimensions

A(: , 1) = 1 2 3 4 A(: , 2) = 10 20 30 40 A(: , 3) = 100 200 300 400 >> sum(A, 3) ans = 111 222 333 444 >> sum(A, 1) ans(: , 1) = 4 6 ans(: , 2) = 40 60 ans(: , 3) = 400 600 >> sum(A, 2) ans(: , 1) = 3 7 ans(: , 2) = 30 70 ans(: , 3) = 300 700

A(: , 1) = 1 2 3 4 A(: , 2) = 10 20 30 40 A(: , 3) = 100 200 300 400 >> min(A, [], 3) ans = 1 2 3 4 >> min(A, [], 1) ans(: , 1) = 1 2 ans(: , 2) = 10 20 ans(: , 3) = 100 200 >> min(A, [], 2) ans(: , 1) = 1 3 ans(: , 2) = 10 30 ans(: , 3) = 100 300

A(: , 1) = 1 2 3 4 A(: , 2) = 10 20 30 40 A(: , 3) = 100 200 300 400 column-order across pages >> A>20 & A<300 ans(: , 1) = 0 0 ans(: , 2) = 0 0 1 1 ans(: , 3) = 1 1 0 0 >> k = find(A>20 & A<300) k = 6 8 9 11

Symbolic Math Toolbox • • • Creating and manipulating symbolic variables Factoring and simplifying algebraic expressions Solving symbolic equations Linear algebra operations Performing summation of infinite series Taylor series expansions, limits Differentiation and integration Solving differential equations Fourier, Laplace, Z-transforms and inverses Variable precision arithmetic

MATLAB Data Classes Character Logical Numeric Integer signed unsigned Symbolic Cell Structure Floating Point single precision double precision More Classes Java classes user-defined classes function handles Symbolic data types are created with sym, syms

>> syms x y z real >> syms x y z positive >> x = sym('x'); >> f = x^6 -1 f = x^6 - 1 >> g = x^3 +1 g = x^3 + 1 >> >> help symbolic doc syms >> sym(sqrt(2)) ans = 2^(1/2) >> sqrt(sym(2)) ans = 2^(1/2) >> double(ans) ans = 1. 4142

>> a = [0, pi/6, pi/4, pi/2, pi]; >> cos(a) ans = 1 0. 8660 0. 7071 >> c = cos(sym(a)) c = [ 1, 3^(1/2)/2, 2^(1/2)/2, >> symdisp(c) display symbolic expression using La. Te. X – on sakai 0 -1 0, -1] >> pretty(c) +-+ | 1/2 | | 3 2 | | 1, ----, 0, -1 | | 2 2 | +-+

>> f = x^6 -1; >> g = x^3 +1; >> r = f/g % x^6 -1 = (x^3 -1)*(x^3+1) r = (x^6 - 1)/(x^3 + 1) >> pretty(r) 6 x - 1 ------3 x + 1 >> symdisp(r) % pretty print

Functions for factoring and simplification of algebraic expressions: simplify expand factor collect simple numden subs - Simplify Expand Factor Collect Search for shortest form Numerator and denominator Symbolic substitution

factor >> simplify(r) ans = x^3 - 1 >> factor(simplify(r)) ans = (x - 1)*(x^2 + x + 1) % x^3 - 1 >> factor(g) ans = (x + 1)*(x^2 - x + 1) % x^3 + 1 >> factor(f) % x^6 - 1 ans = (x - 1)*(x + 1)*(x^2 + x + 1)*(x^2 - x + 1)

>> syms x a b; >> expand((x+1)*(x+2)) ans = x^2 + 3*x + 2 >> expand(exp(a+b)) ans = exp(a)*exp(b) >> expand(cos(a+b)) ans = cos(a)*cos(b) - sin(a)*sin(b) >> expand(cosh(a+b)) ans = sinh(a)*sinh(b) + cosh(a)*cosh(b) expand

>> A = [sin(2*x), sin(3*x) cos(2*x), cos(3*x)] A = [ sin(2*x), sin(3*x)] [ cos(2*x), cos(3*x)] expand >> B = expand(A) B = [2*cos(x)*sin(x), 3*cos(x)^2*sin(x)-sin(x)^3] [cos(x)^2 -sin(x)^2, cos(x)^3 -3*cos(x)*sin(x)^2] >> expand((a+b)*(a^2+2*a*b+b^2)) ans = a^3 + 3*a^2*b + 3*a*b^2 + b^3

>> collect((x+1)*(x+2)) ans = x^2 + 3*x + 2 collect >> collect((a+b)*(a^2 + 2*a*b + b^2), a) ans = a^3 + (3*b)*a^2 + (3*b^2)*a + b^3 >> collect((a+b)*(a^2 + 2*a*b + b^2), b) ans = b^3 + (3*a)*b^2 + (3*a^2)*b + a^3 >> factor(a^3 + 3*a^2*b + 3*a*b^2 + b^3) ans = (a + b)^3

simplify >> simplify(a^3 + 3*a^2*b + 3*a*b^2 + b^3) ans = (a + b)^3 >> simplify(cos(b)*sin(a) - cos(a)*sin(b)) ans = sin(a - b) B = [2*cos(x)*sin(x), 3*cos(x)^2*sin(x)-sin(x)^3] [cos(x)^2 -sin(x)^2, cos(x)^3 -3*cos(x)*sin(x)^2] >> simplify(B) ans = [ sin(2*x), sin(3*x)] [ cos(2*x), cos(3*x)]

numden >> [num, den] = numden(1 + 2*x + 3/(x^2+5)) num = 2*x^3 + x^2 + 10*x + 8 den = x^2 + 5 >> symdisp(1 + 2*x + 3/(x^2+5)) >> symdisp(num/den)

>> syms x a b >> y = a*x+b y = b + a*x x, a, b class sym subs >> y 1 = subs(y, a, 3) y 1 = b + 3*x >> y 2 = subs(y, b, 5) y 2 = a*x + 5 >> y 3 = subs(y, {a, b}, {3, 5}) y 3 = a, b taken from workspace 3*x + 5 >> a=2; b=4; y 4 = subs(y) y 4 = a, b class double 2*x + 4

Functions for solving algebraic and differential equations: solve dsolve finverse compose - solve algebraic equations. solve differential equations. Functional inverse. Functional composition.

Solving symbolic equations >> syms x a b c; >> f = a*x^2 + b*x + c; >> xsol = solve(f) xsol = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) -(b - (b^2 - 4*a*c)^(1/2))/(2*a) >> xsol(1), xsol(2) ans = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) ans = -(b - (b^2 - 4*a*c)^(1/2))/(2*a) symdisp(xsol(1));

>> g = a*(x-xsol(1))*(x-xsol(2)) g = a*(x + (b^2 - 4*a*c)^(1/2))/(2*a))*(x + (b - (b^2 - 4*a*c)^(1/2))/(2*a)) >> simplify(g) ans = a*x^2 + b*x + c >> a*xsol(1)^2 + b*xsol(1) + c ans = c + (b^2 - 4*a*c)^(1/2))^2/(4*a) (b*(b + (b^2 - 4*a*c)^(1/2)))/(2*a) >> simplify(a*xsol(1)^2 + b*xsol(1) + c) ans = 0

some variations >> >> xsol = = solving equations solve('a*x^2 + b*x + c'); solve('a*x^2 + b*x + c=0'); solve('a*x^2 + b*x + c', 'x'); solve(a*x^2 + b*x + c, x); >> syms z; >> xsol = solve(a*z^2 + b*z + c) xsol = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) -(b - (b^2 - 4*a*c)^(1/2))/(2*a) >> bsol = solve(a*x^2 + b*x + c, b) bsol = -(a*x^2 + c)/x

>> syms a b p u w y z >> solve(a+b+p), solve(a+b+w) ans = - a - b how does it know >> solve(u+w+z) which variable ans = to solve for? - u - z alphabetically >> solve(w+y+z) closest to ‘x’ ans = - w - z >> syms x a b c; >> f = a*x^2 + b*x + c; >> xsol = solve(f) xsol = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) -(b - (b^2 - 4*a*c)^(1/2))/(2*a) solving equations

>> f 1 = subs(f, {a, b, c}, {1, 1, -1}) f 1 = x^2 + x – 1 solving equations >> x 1 = solve(f 1) x 1 = 5^(1/2)/2 - 1/2 - 5^(1/2)/2 – 1/2 x 1 class sym >> subs(xsol, {a, b, c}, {1, 1, -1}) ans = -1. 6180 class double 0. 6180 >> phi = simplify(1. /x 1) phi = golden ratio 5^(1/2)/2 + 1/2 - 5^(1/2)/2

Fibonacci numbers, F(n) = F(n-1)+F(n-2) = [0, 1, 1, 2, 3, 5, 8, 13, …] >> syms n >> F = (phi(1)^n – phi(2)^n)/(phi(1)-phi(2)); >> simplify(F - subs(F, n, n-1) - subs(F, n, n-2)) ans = 0 >> limit(F/subs(F, n, n-1), n, inf) ans = 2/(5^(1/2) - 1) >> simplify(limit(F/subs(F, n, n-1), n, inf)) ans = 5^(1/2)/2 + 1/2 golden ratio

solving equations >> eq 1 = 'x^2 + x -1=0'; >> solve(eq 1) ans = 5^(1/2)/2 - 1/2 - 5^(1/2)/2 - 1/2 >> eq 2 = '1/x = x/(1 -x)'; >> solve(eq 2) ans = 5^(1/2)/2 - 1/2 - 5^(1/2)/2 - 1/2 >> solve('1/x = x/(1 -x)');

solving equations >> eq 1 = 'x^2 + y - 1=0'; >> eq 2 = 'x+y=0'; >> [x, y] = solve(eq 1, eq 2) x = 5^(1/2)/2 + 1/2 - 5^(1/2)/2 y = - 5^(1/2)/2 - 1/2 5^(1/2)/2 – 1/2 >> eval(x) ans = 1. 6180 -0. 6180 % or, double(x)

solving equations >> [x, y] = solve('x+y=1', 'x^2+y^2=1'); >> [x, y] ans = [ 1, 0] [ 0, 1] >> S = solve('x+y=1', 'x^2+y^2=1') S = x: [2 x 1 sym] returned as a y: [2 x 1 sym] structure of syms >> [S. x, S. y] ans = [ 1, 0] [ 0, 1]

>> A = [ 5 1 3 0 1 4 1 1 -1 2 6 -2 1 -1 1 4]; >> A = sym(A) A = [ 5, 1, 3, 0] [ 1, 4, 1, 1] [ -1, 2, 6, -2] [ 1, -1, 1, 4] >> inv(A) ans = [ 53/274, [ -21/548, [ 13/548, [ -35/548, linear equations from homework-9 -2/137, -11/274] 34/137, -3/274, -37/548] -8/137, 41/274, 49/548] 11/137, -5/274, 121/548]

>> b = [12; -3; 11; 10]; >> x = inv(A)*b x = 1 -2 3 1 >> class(x) ans = sym % or, Ab linear equations

>> f = 5*x^4 - 2*x^3 + x^2 + 4*x + 3 polynomials >> p = sym 2 poly(f) p = 5 -2 1 4 >> poly 2 sym(p) ans = 5*x^4 - 2*x^3 + x^2 + 4*x + 3 3 >> poly 2 sym(p, 't') ans = 5*t^4 - 2*t^3 + t^2 + 4*t + 3 poly 2 sym 2 poly coeffs quorem roots poly

>> [q, mono] = coeffs(f) q = [ 5, -2, 1, 4, 3] mono = [ x^4, x^3, x^2, x, 1] >> p = sym 2 poly(f) p = 5 -2 1 >> z = roots(p) z = 0. 7393 + 0. 8967 i 0. 7393 - 0. 8967 i -0. 5393 + 0. 3916 i -0. 5393 - 0. 3916 i polynomials 4 3 poly 2 sym 2 poly coeffs quorem roots poly

>> p = sym 2 poly(f) p = 5 -2 1 4 3 polynomials >> z = roots(p) z = 0. 7393 + 0. 8967 i 0. 7393 - 0. 8967 i -0. 5393 + 0. 3916 i -0. 5393 - 0. 3916 i >> P = poly(z) P = 1. 0 -0. 4 >> 5*P ans = 5 -2 poly 2 sym 2 poly coeffs quorem roots poly 0. 2 1 0. 8 4 0. 6 3

>> f = 5*x^4 - 2*x^3 + x^2 + 4*x + 3; >> x 1 = linspace(-1, 1, 201); >> f 1 = subs(f, x 1); >> plot(x 1, f 1, 'b'); >> xlabel('x'); >> ylabel('f(x)'); % or, f 1 = subs(f, x, x 1);

symsum >> syms n >> symsum(1/n^2, n, 1, inf) ans = pi^2/6 >> symsum(1/n^4, n, 1, inf) ans = pi^4/90

>> syms x n N symsum >> symsum(x^n, n, 0, N-1) piecewise([x = 1, N], [x <> 1, . . . (x^N - 1)/(x - 1)]) finite & infinite geometric series >> symsum(x^n, n, 0, inf) piecewise([1 <= x, Inf], . . . [abs(x) < 1, -1/(x - 1)]) >> symsum(x^n/sym('n!'), n, 0, inf) ans = exp(x) >> symsum(n^2, n, 1, N) ans = (N*(2*N + 1)*(N + 1))/6

>> syms x Taylor series >> taylor(exp(x)) ans = x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1 >> taylor(exp(x), 4) ans = x^3/6 + x^2/2 + x + 1 >> taylor(sin(x), 4) ans = x - x^3/6 >> taylor(sin(x)/x) ans = x^4/120 - x^2/6 + 1

>> syms x limits >> limit(exp(-x), x, inf) ans = 0 >> limit(sin(x)/x, x, 0) ans = 1 >> limit(cos(pi*cos(x)/2)/x, x, 0) ans = 0 >> taylor(cos(pi*cos(x)/2)/x, 2) ans = (pi*x)/4

>> syms x simple calculus >> diff(x^2) ans = 2*x >> int(x^2) ans = x^3/3 differentiation integration >> diff(x^3) ans = 3*x^2 >> int(cos(x)) ans = sin(x) >> diff(cos(x)) ans -sin(x) >> int(cos(x)^2, 0, pi) ans = pi/2

acceleration with air drag dsolve terminal velocity and time constant solve with initial velocity v(0) = v 0 exact solution

>> syms v t va ta v 0 D = derivative dsolve >> diffeq = 'Dv=va/ta*(1 -v^2/va^2)'; >> v = dsolve(diffeq, 'v(0)=v 0') v = -va*tan(-va*(- t/(ta*va) + (atan((v 0*i)/va)*i)*i >> v = simplify(v) v = va^2/v 0 + (va*(v 0^2 - va^2))/(v 0*(va + v 0*tanh(t/ta))) >> [num, den] = numden(v);

dsolve >> v = num/den v = (tanh(t/ta)*va^2 + v 0*va)/(va + v 0*tanh(t/ta)) >> symdisp(v); >> u = va*tanh(t/ta + atanh(v 0/va)); >> simplify(v-u) ans = 0

>> subs(v, t, 0) ans = V 0 >> >> >> dsolve t 1 = linspace(0, 301); v 1 = subs(v, {va, ta, v 0, t}, {30, 10, 0, t 1}); plot(t 1, v 1, 'b'); xlabel('t'); ylabel('v(t)');

v 0 = 0; va = 30; ta = 10; tspan = [0, 30]; vdot = @(t, v) va/ta * (1 - v. ^2/va^2); [t 2, v 2] = ode 45(vdot, tspan, v 0); plot(t 2, v 2, 'r-') xlabel('t'); ylabel('v(t)'); compare exact and numerical solutions using ode 45 dsolve

compare exact and numerical solutions using forward Euler method, which replaces derivatives by forward differences dsolve

Tspan = 30; N = 60; T = Tspan/N; tn = 0: T: Tspan; % tn = 0: 0. 5: 30 vn(1) = v 0; dsolve for n=1: N, vn(n+1) = vn(n) + T*va/ta*(1 -vn(n)^2/va^2); end plot(t 1, v 1, 'b'); hold on; plot(tn, vn, 'r--') compare exact and numerical solutions using forward Euler