1. Rutgers University School of Engineering Spring :440:127 - Introduction to Computers for Engineers Sophocles J. Orfanidis ECE Department
week 13

2. Weekly Topics
Week Basics – variables, arrays, matrices, plotting (ch. 2 & 3)
Week Basics – operators, functions, strings, cells (ch. 2,3,11)
Week Matrices (ch. 4)
Week Plotting – 2D and 3D plots (ch. 5)
Week User-defined functions (ch. 6)
Week Input-output formatting – fprintf, fscanf (ch. 7) – Exam 1
Week Relational & logical operators, if, switch statements (ch. 8)
Week For-loops, while-loops (ch. 9)
Week 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

3. Strings, Structures, Cells, Symbolic Math
characters and strings
concatenating strings
using num2str
comparing strings with strcmp
structure arrays
converting structures to cells
cell arrays
cell vs. content indexing
varargin, varargout
multi-dimensional arrays
symbolic math

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

5. Characters and Strings
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
>> c = 'A'
c = A
>> x = double(c)
x =
65 % ASCII code for 'A'
>> char(x)
ans =
>> class(c)
char
char() creates a character string
>> doc char
>> doc class

6. s is a row vector of 8 characters
>> s = 'ABC DEFG'
s =
ABC DEFG
>> x = double(s)
x =
>> char(x)
ans =
>> size(s)
>> class(s)
char
ASCII codes
convert ASCII codes to characters
s is a row vector of 8 characters
>> s(2), s(3:5)
ans = B
C D

7. Concatenating Strings
s = ['Albert', 'Einstein']
s =
AlbertEinstein
>> s = ['Albert', ' Einstein']
Albert Einstein
>> s = ['Albert ', 'Einstein']
>> size(s)
ans =
preserve leading and trailing spaces
>> doc strcat
>> doc strvcat
>> doc num2str
>> doc strcmp
>> doc findstr

8. Concatenating Strings
s = strcat('Albert ', 'Einstein')
s =
AlbertEinstein
>> s = strcat('Albert', ' Einstein')
Albert Einstein
>> fmt = strcat(repmat('%8.3f ',1,6),'\n')
fmt =
%8.3f %8.3f %8.3f %8.3f %8.3f %8.3f\n
strcat strips trailing spaces
but not leading spaces
use repmat to make up long format strings for use with fprintf

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

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

11. num2str a = [143.87, -0.0000325, -7545]'; >> s = num2str(a) s =
-3.25e-005
-7545
>> s = num2str(a,4)
143.9
>> s = num2str(a, '%12.6f')
num2str
s = num2str(A)
s = num2str(A, precision)
s = num2str(A, format)
max number of digits
format spec

12. num2str
a = [143.87, , -7545]';
>> s = num2str(a, '%10.5E')
s =
E+002
E-005
E+003
b = char('A', 'BB', 'CCC');
>> disp([b, repmat(' ',3,1), s])
A E+002
BB E-005
CCC E+003

13. num2str Labeling and saving multiple plots with saved as
t = linspace(0,5,101); w = [2,5,8,10];
Y = sin(w'*t); % 4x101 matrix
s = char('b-', 'r-', 'm-', 'g-');
for i=1:4,
figure; plot(t,Y(i,:), s(i,:));
title(['sin(', num2str(w(i)), 't)']);
print('-depsc', ['file',num2str(i),'.eps']);
end
Labeling and saving
multiple plots with
num2str
saved as
file1.eps, file2.eps
file3.eps, file4.eps

14. Comparing Strings
Strings are arrays of characters, so the condition s1==s2 requires both s1 and s2 to have the same length
>> s1 = 'short'; s2 = 'shore';
>> s1==s1
ans =
>> s1==s2
>> s1 = 'short'; s2 = 'long';
??? Error using ==> eq
Matrix dimensions must agree.

15. strings of unequal length, and get a binary decision
Comparing Strings
Use strcmp to compare
strings of unequal length,
and get a binary decision
>> s1 = 'short'; s2 = 'shore';
>> strcmp(s1,s1)
ans = 1
>> strcmp(s1,s2)
>> s1 = 'short'; s2 = 'long';
>> doc strcmp
>> doc strcmpi
case-insensitive

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

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

18. Structure Arrays add two more students with partially defined fields,
rest of fields are still empty
structure array index
student(2).name = 'Twitter, T.';
student(3).id = ;
>> student
student =
1x3 struct array with fields:
name id
exams
grades

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

20. 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:
exams: [70 80]
grades: []

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

22. s.a = [1 2; 3 4];
s.b = {'a', 'bb'; 'ccc', 'dddd'};
>> s
s =
a: [2x2 double]
b: {2x2 cell}
>> s.a
ans =
>> s.a(2,2) 4
>> s.b
ans =
'a' 'bb'
'ccc' 'dddd'
>> s.b(2,1), s.b{2,1}
'ccc'
ccc
cell vs.
content
indexing

23. Nested Structures student.name = 'Apple, A.'; student.id = 12345;
student.work.exams = [85, 87, 90];
student.work.grades = {'B+', 'B+', 'A'};
>> student
student =
name: 'Apple, A.'
id: 12345
work: [1x1 struct]
>> student.work % sub-structure
ans =
exams: [ ]
grades: {'B+' 'B+' 'A'}

24. Stucture Functions struct - create structure
fieldnames - get structure field names
isstruct test if a structure
isfield test if a field
rmfield remove a structure field
struct2cell - convert structure to cell array
>> C = struct2cell(student)
C =
'Apple, A.'
[ ]
[1x1 struct]
>> C{3}
ans =
exams: [ ]
grades: {'B+' 'B+' 'A'}
content indexing

25. Like structures, cell arrays are containers
of all kinds of data: vectors, matrices, strings, structures, other cell arrays, functions.
A cell is created by putting different types of
objects in curly brackets { }, e.g.,
c = {A, B, C, D}; % 1x4 cell
c = {A; B; C; D}; % 4x1 cell
c = {A, B; C, D}; % 2x2 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 Arrays
cell vs.
content
indexing

26. A = {'Apple'; 'IBM'; 'Microsoft'}; % cells
B = [1 2; 3 4]; % matrix
C x.^2 + 1; % function
D = [ ]; % row
c = {A,B;C,D} % define 2x2 cell array
c =
{3x1 cell} [2x2 double]
@(x)x.^2+1 [1x5 double]
>> cellplot(c);

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

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

29. A = {'Apple'; 'IBM'; 'Microsoft'}; % cells
B = [1 2; 3 4]; % matrix
C x.^2 + 1; % function
D = [ ]; % row
c = {A,B;C,D} % define 2x2 cell array
c =
{3x1 cell} [2x2 double]
@(x)x.^2+1 [1x5 double]
>> cellplot(c);

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

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

32. cell indexing ( ) content indexing { } cell cell contents
ans =
[1x5 double]
>> class(c(2,2))
cell
>> c{2,2}
>> class(c{2,2})
double
cell
cell contents

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

34. >> d = c;
>> % d(1,3) = {[4 5 6]'}; % define as cell
>> d{1,3} = [4,5,6]' % define content
d =
{3x1 cell} [2x2 double] [3x1 double]
@(x)x.^2+1 [1x5 double] []
>> d(2,3)
ans =
{[]}
>> d{2,3} []
>> cellplot(d);

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

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

37. num2cell cell2mat >> A = [1 2; 3 4]; >> C = num2cell(A)
cell2mat converts numerical cell arrays to numbers num2cell converts numerical array to cell array
>> A = [1 2; 3 4];
>> C = num2cell(A)
C =
[1] [2]
[3] [4]
>> B = cell2mat(C)
B =

38. num2cell cell2mat >> A = [1,2; 3,4]; % 2x2 matrix
>> S = {'a', 'bb'; 'ccc', 'dddd'}; % 2x2 cell
S =
'a' 'bb'
'ccc' 'dddd'
>> [A,S] % can’t mix numbers and cells
??? Error using ==> horzcat
CAT arguments dimensions are not consistent.
>> [num2cell(A),S] % construct 2x4 cell
ans =
[1] [2] 'a' 'bb'
[3] [4] 'ccc' 'dddd'

39. using num2cell with fprintf
S = {'January', 'April', 'July', 'October'};
T = [30, 60, 80, 70];
C = [S;T] % can’t mix numbers and cells
??? Error using ==> vertcat
CAT arguments dimensions
are not consistent.
C = [S; num2cell(T)] % make all into cells
C =
'January' 'April' 'July' 'October'
[ ] [ 60] [ 80] [ ]
fprintf(' %-7s %5.2f\n', C{:});
January
April
July
October
>> class(C)
ans =
cell
>> C(:)
'January'
[30]
'April'
[60]
'July'
[80]
'October'
[70]
read content column-wise
print it row-wise

40. num2cell fprintf column vectors transposed content indexing
M = {'January'; 'April'; 'July'; 'October'};
T1 = [30; 60; 80; 70];
T2 = [-10; 40; 60; 50];
C = [M, num2cell([T1,T2])]'
% C =
% 'January' 'April' 'July' 'October'
% [ ] [ 60] [ 80] [ ]
% [ ] [ 40] [ 60] [ ]
fprintf(' T T2\n')
fprintf(' \n')
fprintf('%-7s %6.2f %6.2f\n',C{:})
% fprintf('%-7s %6.2f %6.2f\n',C{:,:})
T T2
January
April
July
October
num2cell
fprintf
column vectors
transposed
>> C(:)
ans =
'January'
[ 30]
[-10]
'April'
[ 60]
[ 40]
'July'
[ 80]
'October'
[ 70]
[ 50]
content
indexing

41. varargin
varargout
varargin,varargout are cell arrays that allow the passing a variable number of function inputs & outputs
% [x,y,vx,vy] = trajectory(t,v0,th0,h0,g)
function [varargout] = trajectory(t,v0,varargin)
Nin = nargin-2; % number of varargin inputs
if Nin==0, th0=90; h0=0; g=9.81; end
if Nin==1, th0=varargin{1}; h0=0; g=9.81; end
if Nin==2, th0=varargin{1}; ...
h0=varargin{2}; g=9.81; end
if Nin==3, th0=varargin{1}; ...
h0=varargin{2}; g=varargin{3}; end
continues

42. th0 = th0 * pi/180; % convert to radians
x = v0 * cos(th0) * t;
y = h0 + v0 * sin(th0) * t - 1/2 * g * t.^2;
vx = v0 * cos(th0);
vy = v0 * sin(th0) - g * t;
if nargout==1; varargout{1} = x; end
if nargout==2; varargout{1} = x; ...
varargout{2} = y; end
if nargout==3; varargout{1} = x;
varargout{2} = y; ...
varargout{3} = vx; end
if nargout==4; varargout{1} = x; ...
varargout{3} = vx; ...
varargout{4} = vy; end

43. 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 4D array is a collection of 3D arrays of the same size.
applications in
video processing

44. pages can operate along the i,j,p dimensions >> a = [1 2; 3 4];
>> A(:,:,1) = a;
>> A(:,:,2) = 10*a;
>> A(:,:,3) = 100*a;
>> A
A(:,:,1) =
A(:,:,2) =
A(:,:,3) =
pages
sum, min, max
can operate along the
i,j,p dimensions

45. >> sum(A,1)
ans(:,:,1) =
ans(:,:,2) =
ans(:,:,3) =
>> sum(A,2) 3 7 30 70
300
700
A(:,:,1) =
A(:,:,2) =
A(:,:,3) =
>> sum(A,3)
ans =

46. >> min(A,[],1)
ans(:,:,1) =
ans(:,:,2) =
ans(:,:,3) =
>> min(A,[],2) 1 3 10 30
100
300
A(:,:,1) =
A(:,:,2) =
A(:,:,3) =
>> min(A,[],3)
ans =

47. column-order across pages >> A>20 & A<300 ans(:,:,1) = 0 0
ans(:,:,2) =
ans(:,:,3) =
>> k = find(A>20 & A<300)
k = 6 8 9 11
A(:,:,1) =
A(:,:,2) =
A(:,:,3) =
column-order
across pages

48. 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

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

50. >> help symbolic >> doc symbolic >> doc sym
>> syms x y z
>> syms x y z real
>> syms x y z positive
>> x = sym('x');
>> help symbolic
>> doc symbolic
>> doc sym
>> doc syms
>> sym(sqrt(2))
ans =
2^(1/2)
>> sqrt(sym(2))
>> double(ans)
1.4142
>> f = x^6-1
f =
x^6 - 1
>> g = x^3 +1
g =
x^3 + 1

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

52. >> 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) % pretty print 6
x - 1 3
x + 1
>> symdisp(r)

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

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

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

56. expand >> A = [sin(2*x), sin(3*x) cos(2*x), cos(3*x)] A =
>> 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
expand

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

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

59. 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)

60. subs >> syms x a b x,a,b class sym >> y = a*x+b y =
b + a*x
>> y1 = subs(y,a,3)
y1 =
b + 3*x
>> y2 = subs(y,b,5)
y2 =
a*x + 5
>> y3 = subs(y,{a,b},{3,5})
y3 =
3*x + 5
>> a=2; b=4; y4 = subs(y)
y4 =
2*x + 4
x,a,b class sym
subs
a,b taken from workspace
a,b class double

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

62. 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 =
symdisp(xsol(1));

63. >> g = a*(x-xsol(1))*(x-xsol(2))
a*(x + (b + (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
c + (b + (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)

64. solving equations some variations
>> xsol = solve('a*x^2 + b*x + c');
>> xsol = solve('a*x^2 + b*x + c=0');
>> xsol = solve('a*x^2 + b*x + c','x');
>> xsol = 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

65. solving equations >> syms a b p u w y z
>> solve(a+b+p), solve(a+b+w)
ans =
- a - b
>> solve(u+w+z)
- u - z
>> solve(w+y+z)
- w - z
solving equations
how does it know
which variable
to solve for?
alphabetically
closest to ‘x’
>> 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)

66. solving equations class double golden ratio
>> f1 = subs(f,{a,b,c},{1,1,-1})
f1 =
x^2 + x – 1
>> x1 = solve(f1)
x1 =
5^(1/2)/2 - 1/2
- 5^(1/2)/2 – 1/2
>> subs(xsol,{a,b,c},{1,1,-1})
ans =
0.6180
>> phi = simplify(1./x1)
phi =
5^(1/2)/2 + 1/2
1/2 - 5^(1/2)/2
solving equations
x1 class sym
class double
golden ratio

67. 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 =
>> limit(F/subs(F,n,n-1), n, inf)
2/(5^(1/2) - 1)
>> simplify(limit(F/subs(F,n,n-1), n,inf))
5^(1/2)/2 + 1/2
golden ratio

68. solving equations >> eq1 = 'x^2 + x -1=0'; >> solve(eq1)
ans =
5^(1/2)/2 - 1/2
- 5^(1/2)/2 - 1/2
>> eq2 = '1/x = x/(1-x)';
>> solve(eq2)
>> solve('1/x = x/(1-x)');

69. solving equations >> eq1 = 'x^2 + y - 1=0';
>> [x,y] = solve(eq1,eq2)
x =
5^(1/2)/2 + 1/2
1/2 - 5^(1/2)/2
y =
- 5^(1/2)/2 - 1/2
5^(1/2)/2 – 1/2
>> eval(x) % or, double(x)
ans =
1.6180

70. solving equations returned as a structure of syms
>> [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: [2x1 sym]
y: [2x1 sym]
>> [S.x, S.y]
returned as a
structure of syms

71. linear equations >> 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, -2/137, -12/137, -11/274]
[ -21/548, 34/137, -3/274, -37/548]
[ 13/548, -8/137, 41/274, 49/548]
[ -35/548, 11/137, -5/274, 121/548]
from homework-9

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

73. polynomials >> f = 5*x^4 - 2*x^3 + x^2 + 4*x + 3 f =
>> p = sym2poly(f)
p =
>> poly2sym(p)
ans =
>> poly2sym(p,'t')
5*t^4 - 2*t^3 + t^2 + 4*t + 3
polynomials
poly2sym
sym2poly
coeffs
quorem
roots
poly

74. polynomials >> [q,mono] = coeffs(f) q = [ 5, -2, 1, 4, 3] mono =
[ x^4, x^3, x^2, x, 1]
>> p = sym2poly(f)
p =
>> z = roots(p)
z = i i i i
polynomials
poly2sym
sym2poly
coeffs
quorem
roots
poly

75. polynomials >> p = sym2poly(f) p = 5 -2 1 4 3
>> z = roots(p)
z = i i i i
>> P = poly(z)
P =
>> 5*P
ans =
polynomials
poly2sym
sym2poly
coeffs
quorem
roots
poly

76. >> f = 5*x^4 - 2*x^3 + x^2 + 4*x + 3;
>> x1 = linspace(-1,1,201);
>> f1 = subs(f,x1); % or, f1 = subs(f,x,x1);
>> plot(x1,f1,'b');
>> xlabel('x');
>> ylabel('f(x)');

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

78. symsum >> syms x n N >> symsum(x^n, n, 0, N-1)
piecewise([x = 1, N], [x <> 1,...
(x^N - 1)/(x - 1)])
>> 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)
(N*(2*N + 1)*(N + 1))/6
finite & infinite
geometric series

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

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

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

82. dsolve acceleration with air drag terminal velocity and time constant
solve with initial
velocity v(0) = v0
exact solution

83. dsolve D = derivative >> syms v t va ta v0
>> diffeq = 'Dv=va/ta*(1-v^2/va^2)';
>> v = dsolve(diffeq, 'v(0)=v0')
v =
-va*tan(-va*(- t/(ta*va) +
(atan((v0*i)/va)*i)/va)*i)*i
>> v = simplify(v)
va^2/v0 +
(va*(v0^2 - va^2))/(v0*(va + v0*tanh(t/ta)))
>> [num,den] = numden(v);

84. dsolve >> v = num/den v =
(tanh(t/ta)*va^2 + v0*va)/(va + v0*tanh(t/ta))
>> symdisp(v);
>> u = va*tanh(t/ta + atanh(v0/va));
>> simplify(v-u)
ans =

85. dsolve >> subs(v,t,0) ans = V0 >> t1 = linspace(0,30,301);
>> v1 = subs(v, {va,ta,v0,t}, {30,10,0,t1});
>> plot(t1,v1,'b');
>> xlabel('t');
>> ylabel('v(t)');

86. dsolve v0 = 0; va = 30; ta = 10; tspan = [0,30];
vdot va/ta * (1 - v.^2/va^2);
[t2,v2] = ode45(vdot, tspan, v0);
plot(t2,v2,'r-')
xlabel('t');
ylabel('v(t)');
compare exact and
numerical solutions
using ode45

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

88. dsolve Tspan = 30; N = 60; T = Tspan/N;
tn = 0:T:Tspan; % tn = 0:0.5:30
vn(1) = v0;
for n=1:N,
vn(n+1) = vn(n) + T*va/ta*(1-vn(n)^2/va^2);
end
plot(t1,v1,'b');
hold on;
plot(tn,vn,'r--')
compare exact and
numerical solutions
using forward Euler