2013. 03. 12
Brief Guide to
msharpmath (cemmsharp)
Lecture by
www.msharpmath.com
Prof. Charn-Jung Kim
Seoul National University
School of Mechanical and Aerospace Engineering
Seoul National University
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Why msharpmath ?
Need a paradigm shift :
grammar-based to human thinking
Find a root of
MATLAB
grammar-based
function f = myfun(x)
f = x^2 – 3*x + 2
>> fzero( @myfun, 0 )
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x 2  3x  2
M#Math
human-thinking
#> solve .x ( x^2 = 3*x – 2 );
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How to download msharpmath
• HomePage : www.msharpmath.com
Click here
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msharpmath Screen
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Mathematics Calculator
#> 3 + 4 * 5
ans =
23
#> ans/1000 + ans*100
ans =
2300.023
cursor in msharpmath
#> 3 + 4 * 5
default variable name
for results
Variable “ans” can be
used in subsequent
commands
ans =
23
#> ans / 1000 + ans * 100
ans =
2300.023
Variable “ans” is changed to a new value
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■ Unit conversion
x.f // Fahrenheit to other temperatures
x.c // Celsius to other temperatures
x.ft // feet in meter
x.acre// acre in square meter
and many more
#> 77.f
25 [C]
77 F
536.67 R
298.15 K
ans =
298.15
#> 1.ft * 1
ans =
0.3048
#> 1.acre * 1
ans =
4046.8564
expression-look-alike !!!
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■ Root finding
solve.x{a} ( f(x)=g(x) ) // a root of f(x)=g(x) near a
solve.x.y {a,b} ( u(x,y)=0, v(x,y)=0 )
solve.x.y.z {a,b,c} (
u(x,y,z)=0, v(x,y,z)=0, w(x,y,z)=0 )
#> solve.x ( exp(x) = x+3 )
ans =
1.5052415
#> solve.x ( exp(x) = x+3 ).span(-10,10)
ans =
[
-2.94753 -5.66165e-008 ]
[
1.50524 -1.89955e-008 ]
#> solve.a.b ( a+b = 3, a*b = 2 )
ans =
[
2 ]
[
1 ]
expression-look-alike !!!
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■ Integration
int.x(a,b) ( f(x) )
int.x(a,b).y(u(x),v(x)) ( f(x,y) )
int.x(a,b).y(u(x),v(x)).z(p(x,y),q(x,y)) ( f(x,y,z) )
#> int.x(1,2).y(x,2*x+1) ( exp(-x-y)*sin(x+y) )
ans =
2

1
2 x 1
x
-0.0089769673
e  x  y sin( x  y )dydx
expression-look-alike !!!
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double
● data type : double (real numbers)
msharpmath (m#math)
MATLAB
double (real number)
complex
poly (polynomial)
matrix
vertex
signal
image
...
matrix
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real numbers are treated
by 1 x 1 matrix
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Operators for double (1)
double
|a|
-a
!a
// absolute
// unary minus
// return 1 if a=0, otherwise return 0
#> a = -1.7
a =
-1.7
#> |a|
ans =
1.7
#> -a
ans =
1.7
#> !a
ans =
0
#> !0
ans =
1
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Operators for double (2)
double
a
a
a
a
a
a
a
+
*
/
\
^
%
b
b
b
b
b
b
b
//
//
//
//
//
//
//
#> 2 + 3
ans =
#> 2 - 3
ans =
#> 2 * 3
ans =
addition
subtraction
multiplication
right division
left division
power
remainder, equal to .mod(a,b)
5
#> 2 / 3
ans =
0.66666667
-1
#> 2 \ 3
ans =
1.5
6
#> 2 ^ 3
ans =
8
#> 13 % 5
ans =
3
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Operators for double (3)
double
Compound operations
a
a
a
a
a
a
+=
-=
*=
/=
^=
%=
b
b
b
b
b
b
//
//
//
//
//
//
a
a
a
a
a
a
=
=
=
=
=
=
a+b
a-b
a*b
a/b
a^b
a%b
#> a = 3
a =
3
#> a += 4
a =
7
#> a ^= 2
a =
49
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poly
● data type : poly (polynomial)
msharpmath (m#math)
MATLAB
double (real number)
complex
poly (polynomial)
matrix
vertex
signal
image
...
matrix
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polynomials are
treated by functions
such as
polyval(p,x)
polyder(p,x)
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poly
ascending poly
poly( a0, a1, a2, a3, … , an )
[ a0, a1, a2, a3, … , an ] .apoly
p( x)  a0  a1 x  a2 x2  a3 x3 
 an xn
p(x) = a0 + a1 x + a2 x^2 + a3 x^3 + … + an x^n
#> poly( 1,2,3,4,5 )
ans = poly( 1 2 3 4 5 )
= 5x^4 +4x^3 +3x^2 +2x +1
#> [ 1,2,3,4,5 ].apoly
ans = poly( 1 2 3 4 5 )
= 5x^4 +4x^3 +3x^2 +2x +1
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descending poly
poly
[
.[
.[
.[
.[
.[
an, … , a3, a2,
an, … , a3, a2,
a ]
//
a, b ]
//
a, b, c ]
//
a, b, c, d ] //
a1, a0 ]
a1, a0 ]
constant
1st-order
2nd-order
3rd-order
#> .[ 1,2,3,4,5 ]
ans = poly( 5 4 3 2 1 )
= x^4 +2x^3 +3x^2 +4x +5
#> .[ 1,3,2 ]
y = x^2 + 3x + 2
roots
[
-1 ]
[
-2 ]
vertex at ( -1.5, -0.25 )
y_minimum = -0.25
y-axis intersect ( 0, 2 )
symmetry at x = -1.5
directrix at y = -0.5
focus at ( -1.5, 0 )
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.dpoly
Note : a leading dot for
descending polynomial !!!
polynomial
polynomial, ax+b
polynomial, ax^2+bx+c
polynomial, ax^3+bx^2+cx+d
#> .[]
ans = poly( 0)
= 0
#> p = q = r = .[]
p = poly( 0)
= 0
q = poly( 0)
= 0
r = poly( 0)
= 0
.[] is a zero polynomial
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Operations for poly (1)
poly
-p
p’
p~
p.’
p.~
//
//
//
//
//
unary minus
dp(x)/dx,
int_0^x p(x) dx,
p(x)-p(x-1),
p(1)+p(2)+…+p(n),
#> .[1,0,0,0]'
ans = poly( 0
= 3x^2
0
#> .[1,0,0,0].'
ans = poly( 1 -3
= 3x^2 -3x +1
#> .[1,0,0,0]~
ans = poly( 0
= 0.25x^4
0
derivative
integration
finite difference
cumulative sum
d 3
x  3x 2
dx
3 )
x3  ( x  1)3  3x2  3x  1
3 )
0
0
0.25 )
#> .[1,0,0,0].~
ans = poly( 0 0 0.25 0.5 0.25 )
= 0.25x^4 +0.5x^3 +0.25x^2
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
x
0
x3 dx 
1 4
x
4
13  23  33 
1

 x3   x( x  1) 
2

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Operations for poly (2)
poly
p
p
p
p
q
p
P
+ q
- q
* q
/ q
\ p
% q
%% q
//
//
//
//
//
//
//
addition
subtraction
multiplication
quotient(right division)
quotient(left division)
remainder
synthetic division
p( x)  3x2  4x  5
q ( x)  x  2
#> p = .[3,4,5]
p = poly( 5 4 3 )
= 3x^2 +4x +5
#> q = .[1,2]
q = poly( 2 1 )
= x +2
#> p + q
ans = poly( 7 5 3 )
= 3x^2 +5x +7
#> p / q
ans = poly( -2
= 3x -2
#> p - q
ans = poly( 3 3 3 )
= 3x^2 +3x +3
#> p % q
ans = poly( 9 )
= 9
#> p * q
ans = poly( 10 13 10 3 )
= 3x^3 +10x^2 +13x +10
p( x)  q( x)(3x  2)  9
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Operations for poly (3)
poly
p(x)
p(z)
p(q)
p(A)
p[A]
//
//
//
//
//
double
complex
polynomial
matrix (element-by-element)
matrix polynomial, a0 I+a1 A+a2 A^2 + …
#> p = .[1,0,0]
p = poly( 0
= x^2
0
p ( x)  x 2
1 )
(3)2  9
#> p(3)
ans =
9
#> p(3+1i)
ans = 8 + 6!
(3  i)2  9  6i  1  8  6i
#> p( .[3,4] )
ans = poly( 16 24 9 )
= 9x^2 +24x +16
(3x  4)2  9 x2  24 x  16
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Operations for poly (4)
poly
p(x)
p(z)
p(q)
p(A)
p[A]
//
//
//
//
//
double
complex
polynomial
matrix (element-by-element)
matrix polynomial, a0 I+a1 A+a2 A^2 + …
p ( x)  x 2
#> p( [1,2;3,4] )
ans =
[
1
[
9
#> p[ [1,2;3,4] ]
ans =
[
7
[
15
4 ]
16 ]
10 ]
22 ]
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12
 2
3
1
3

2 2  1 4 


42  9 16 
2
2
1

3
4 

2  1
4  3
2  7

4  15
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22 
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Member functions
poly
p.solve
p.pow(k)
p.shift(k)
p.up(k)
p.newton(u)
//
//
//
//
//
roots
replace x by x^k, p[i*k]=p[i]
replace x by x-k
multiply by x^k, p[i+k]=p[i]
a0+a1(x-u1)+a2(x-u1)(x-u2)+…
#> .[1,2,5].solve
ans =
[
-1 - i 2
[
-1 + i 2
]
]
#> p = .[1,0,0]; p.pow(3);
p = x^2
ans = x^6
p( x)  x2 ,
#> p.shift(2)
ans = = x^2 -4x +4
p( x  2)  ( x  2)2
#> p.up(3)
ans = x^5
p( x) x3  x2 x3
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p( x3 )  ( x3 )2
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poly
exercise for poly
(3x2  5x  7)3  (5x  4 x2 )( x3  6 x2  15x  24)  32( x  17)3
#> .[3,-5,7]^3 - poly(0,5,4)*.[1,-6,15,24] + 32*.[1,-17]^3
ans = poly( -156873 26889 -837 -753 433
= 27x^6 -139x^5 +433x^4 -753x^3 -837x^2
+26889x -156873
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-139
27 )
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complex
● data type : complex
msharpmath (m#math)
MATLAB
double (real number)
complex
poly (polynomial)
matrix
vertex
signal
image
...
matrix
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complex numbers are
treated by 1 x 1 matrix
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Creating complex numbers
complex
1i, 1j, 1!
x+y!
(r+t!).cyl
#> 1i
ans =
// i or j works only after digits 0-9
// postfix ! for pure imaginary part
// r cos(t) + r sin(t) !
‘i’ after digits
0 + 1!
#> 1j
ans =
0 + 1!
#> 1!
ans =
0 + 1!
#> 3+pi!
ans =
3 + 3.14159!
#> (3+pi!).cyl
ans = -3 + 3.67394e-016!
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‘j’ after digits
‘!’ after both digits and
variables
In polar coordinate
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Elements of complex
complex
z = x+y!
|z|
~z
z.x, z.real
z.y, z.imag
z.r, z.abs
z.t, z.arg
z.deg
//
//
//
//
//
//
//
absolute value
conjugate, x-y!
real part, x
imaginary part, x
equal to |z|=sqrt(x*x+y*y)
phase part in radian, tan^-1(y/x)
phase part in degree
#> z = 3+4i
z = 3 + 4!
#> z.r
ans =
5
#> |z|
ans =
#> z.t
ans =
0.92729522
#> z.deg
ans =
53.130102
#> ~z
ans =
#> z.y
ans =
5
3 - 4!
4
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Be cautious with
complex
A function “sqrt” returns
‘double’ for ‘double’,
‘complex’ for ‘complex’
#> sqrt(-1)
ans =
1 
sqrt(-1)
NaN  sqrt(1)
i  1  sqrt(1  0i)
-1.#IND
#> sqrt(-1+0i)
ans = 6.12323e-017 + 1!
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matrix
● data type : matrix
msharpmath (m#math)
MATLAB
double (real number)
complex
poly (polynomial)
matrix
vertex
signal
image
...
matrix
Matrix Laboratory
All the data are
processed based on matrix
 a11
a
A   21


 am1
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a12
a22
am 2
a1n 
a2 n 


amn 
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Matrix by a pair of brackets []
matrix
[]
// 0 x 0 matrix
[ a11,a12,…,a1n; a21,a22,… ; … ; a_m1,…, a_mn ]
// separate rows by a semi-colon
// separate elements by a comma
#> A = [ 1,2,3; 4,5; 6 ]
A =
[
1
2
3 ]
[
4
5
0 ]
[
6
0
0 ]
#> [ A,-A; 2*A ]
ans =
[
1
2
3
[
4
5
0
[
6
0
0
[
2
4
6
[
8
10
0
[
12
0
0
-1
-4
-6
0
0
0
;
,
All zeros are assigned
for undeclared elements
-2
-5
-0
0
0
0
-3
-0
-0
0
0
0
]
]
]
]
]
]
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Matrices can be also
elements of a matrix
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Matrix by a colon operator
matrix
a:b
// [ a,a+1,a+2, … ] if a < b
a:h:b
// [ a,a+h,a+2h, … ]
#> 1:5
ans =
[
1
2
3
4
5 ]
#> 5:1
ans =
[
5
4
3
2
1 ]
1
1.2
1.4
1.6
#> 1 : 0.2 : 2
ans = [
1.8
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// [ a,a-1,a-2, … ] if a > b
2 ]
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Matrix by built-in functions
matrix
.I(m,n=m)
or
.ones(m,n=m)
.zeros(m,n=m)
#> .I(2,3)
ans =
[
1
0
[
0
1
#> .ones(4)
ans =
[
1
1
[
1
1
[
1
1
[
1
1
#> .zeros(A)
ans =
[
0
0
[
0
0
[
0
0
.eye(m,n=m)
equal to eye(2,3)
0 ]
0 ]
equal to ones(4,4)
1
1
1
1
1
1
1
1
]
]
]
]
Same size with A
0 ]
0 ]
0 ]
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Operations for matrices
matrix
+
*
/
\
^
‘
//
//
//
//
//
//
//
addition
subtraction
multiplication
right division, A/B = A*B^-1
left division,
A\B = A^-1*B
power,
A^3 = A*A*A
complex conjugate transpose
#> A = [ 1,2; 3,4 ];
A =
[
1
2 ]
[
3
4 ]
b =
[
5 ]
[
6 ]
b = [ 5; 6 ]
#> A \ b
ans =
[
-4 ]
[ 4.5 ]
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Ax = b,
1
3

2   x1  5

4   x2  6
x = A1b  A \ b
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Operations with scalars
matrix
Scalar is expanded as a matrix
#> A = [ 1,2,3; 4,5,6 ]
A =
[
1
2
3 ]
[
4
5
6 ]
#> A + 5
ans =
[
6
7
[
9
10
8 ]
11 ]
#> A ^ 5
runtime error#90700: matrix not square
#> A = 5
A =
[
5
[
5
5
5
5 ]
5 ]
1
4

 a11
a
 21
 a11
a
 21
2
5
3
1

5

4
6 

a12
a22
a13 
5

a23 
a12
a22
a13  5

a23  5
3  5

6  5
2
5
5
5
5
5
5
5 
5
5
in MATLAB, A = 5 results in a scalar (1 x 1 matrix)
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matrix
.*
./
.\
.^
.‘
//
//
//
//
//
Element-by-element operations
element-by-element
element-by-element
element-by-element
element-by-element
pure transpose
multiplication
right division
left division
power
#> A = [ 1,2; 3,4 ]; B = [ 5,6; 7,8 ]
A =
[
1
2 ]
[
3
4 ]
B =
[
5
6 ]
[
7
8 ]
#> A .* B
ans =
[
5
[
21
12 ]
32 ]
#> A .\ B
ans =
[
5
[ 2.33333
3 ]
2 ]
#> A ./ B
ans =
[
0.2 0.333333 ]
[ 0.428571
0.5 ]
#> B .^ A
ans =
[
5
[
343
36 ]
4096 ]
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Dot and Kronecker Products
matrix
A ** B
A ^^ B
// dot product, .dot(A,B)
// Kronecker product, .kron(A,B)
#> A = [ 1,10,-100; -10,1;1000 ]; B = [ 1,2,3; 4,5,6; 7,8,9 ]
A =
[
1
10 -100 ]
[ -10
1
0 ]
[ 1000
0
0 ]
B =
[
1
2
3 ]
[
4
5
6 ]
#> B ** B
[
7
8
9 ]
ans =
285
#> A ^^
ans
[
1
[
4
[
7
[ -10
[ -40
[ -70
[ 1000
[ 4000
[ 7000
B
=
2
3
5
6
8
9
-20 -30
-50 -60
-80 -90
2000 3000
5000 6000
8000 9000
10
40
70
1
4
7
0
0
0
20
50
80
2
5
8
0
0
0
30 -100 -200 -300 ]
60 -400 -500 -600 ]
90 -700 -800 -900 ]
3
0
0
0 ]
6
0
0
0 ]
9
0
0
0 ]
0
0
0
0 ]
0
0
0
0 ]
0
0
0
0 ]
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Concatenations
matrix
A | B
A _ B
A \_ B
// horizontal concatenation, .horzcat(A,B,C,…)
// vertical concatenation,
.vertcat(A,B,C,…)
// diagonal concatenation,
.diagcat(A,B,C,…)
#> A = [ 1,2,3 ]; B = [ -4,5 ]
A = [
1
2
B = [
-4
5 ]
#> A | B
ans =
[
1
2
#> A _ B
ans =
[
1
[
-4
2
5
3 ]
0 ]
#> A \_ B
ans =
[
1
[
0
2
0
3
0
3 ]
3
0
-4
School of Mechanical and Aerospace Engineering
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-4
5 ]
0 ]
5 ]
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Set operations
matrix
A ++ B
A -- B
A /\ B
// set union
// set difference
// set intersection
#> A = [ 2,3,4,3,4,6 ]; B = [ 1,2,3,2,3,8,8,9 ]
A = [
2
3
4
3
4
6 ]
B = [
1
2
3
2
3
8
8
#> A ++ B
ans =
[
2
3
#> A -- B
ans =
[
4
6 ]
#> B -- A
ans =
[
1
8
[
2
3 ]
4
6
1
8
9 ]
9 ]
9 ]
#> A /\ B
ans =
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Compound assignments
matrix
A +=
A -=
A *=
A /=
A |=
A _=
A\_=
B
B
B
B
B
B
B
//
//
//
//
//
//
//
A
A
A
A
A
A
A
=
=
=
=
=
=
=
A +
A A *
A /
A |
A _
A\_
B
B
B
B = A(B^-1)
B, horizontal concatenation
B, vertical concatenation
B, diagonal concatenation
A .*= B
A ./= B
A .^= B
// A = A .* B
// A = A ./ B
// A = A .^ B
A ++= B
A --= B
A /\= B
// A = A ++ B, set union
// A = A –- B, set difference
// A = A /\ B, set intersection
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Access to elements (1)
matrix
A(i,j), A(k)
A{i,j), A{k}
A[i,j], A[k]
// real element
// imaginary element
// complex element
#> A = [ 1, 3-4! ; -2+5i, 6; 7,-8i ]
A =
[
1
3 - i 4
]
[
-2 + i 5
6
]
[
7
-0 - i 8
]
#> A(2,1); A{2,1}; A[2,1];
ans =
-2
ans =
5
ans = -2 + 5!
This is a unique feature of msharpmath compared
with MATLAB and similar interactive languages
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Access to elements (2)
matrix
A.row(i), A(i,*) // i-th row vector (1 x n matrix)
A.col(j), A(*,j) // j-th column vector (m x 1 matrix)
A.diag(k) // diagonal vector (k=0 where i=j)
#> A = [ 2,5,8; 3,6,9; 4,7,10 ]
A =
[
2
5
8 ]
[
3
6
9 ]
[
4
7
10 ]
#> A.col(3)
ans =
[
8 ]
[
9 ]
[
10 ]
School of Mechanical and Aerospace Engineering
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#> A.diag(0) -= 20
A =
[
-18
5
[
3
-14
[
4
7
8 ]
9 ]
-10 ]
#> A.row(2) *= 100
A =
[
-18
5
[
300 -1400
[
4
7
8 ]
900 ]
-10 ]
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Access to elements (3)
matrix
A.(matrix) // entry of elelements, A.entry(matrix)
#> A = [ 10,40,70,100; 20,50,80,110; 30,60,90,120 ]
A =
[
10
40
70
100 ]
[
20
50
80
110 ]
[
30
60
90
120 ]
#> A.( [2,6,10] )
ans = [
20
#> A.( [2,6,10] ) = 0
A =
[
10
40
70
[
0
50
80
[
30
0
90
60
100 ]
2nd, 6th, and 10th elements
0 ]
110 ]
120 ]
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Access to elements (4)
matrix
A..(rows)(columns) // submatrix, A.sub(rows)(columns)
#> x = 1:6
x = [
1
#> A = [ x+10; x+20;
A =
[
11
12
[
21
22
[
31
32
[
41
42
[
51
52
[
61
62
2
3
4
5
6 ]
x+30; x+40; x+50; x+60 ]
13
23
33
43
53
63
14
24
34
44
54
64
15
25
35
45
55
65
16
26
36
46
56
66
]
]
]
]
]
]
#> A ..( 2,1,[4,3],6) ( 2,[5,4],1 )
ans =
[
22
25
24
21 ]
[
12
15
14
11 ]
[
42
45
44
41 ]
[
32
35
34
31 ]
[
62
65
64
61 ]
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Deleting elements
matrix
A.row(i) = []
// A = A.rowdel(i)
A.col(j) = []
// A = A.coldel(i)
A..(rows)(columns) = [] // A = A.subdel(rows)(columns)
#> A = [ 11,12,13,14; 21,22,23,24; 31,32,33,34 ]
A =
[
11
12
13
14 ]
[
21
22
23
24 ]
[
31
32
33
34 ]
#> A.row(2) = []
A =
[
11
12
[
31
32
13
33
#> A.col(3) = []
A =
[
11
12
[
31
32
14 ]
34 ]
14 ]
34 ]
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Extending matrices (1)
matrix
A.resize(m,n) // new m x n dimension, while copying
A.ext(m,n,s) // extend m rows, n columns and assign s
#> A = [ 1,2,3; 4,5,6
A =
[
1
2
[
4
5
#> A.resize(4,6)
ans =
[
1
2
[
4
5
[
0
0
[
0
0
#> A.ext(3,2,
ans =
[
1
[
4
[
-1
[
-1
[
-1
]
3 ]
6 ]
3
6
0
0
0
0
0
0
0
0
0
0
3
6
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
0
0
0
0
]
]
]
]
#> A.resize(4,2)
ans =
[
1
2
[
4
5
[
0
0
[
0
0
]
]
]
]
-1)
2
5
-1
-1
-1
]
]
]
]
]
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Extending matrices (2)
matrix
A.insrow(i,B) // insert row
A.inscol(j,B) // insert column
A.ins(i,j,B) // insert matrix
#> A = (1:9)._3
A =
[
1
4
[
2
5
[
3
6
7 ]
8 ]
9 ]
#> A.insrow(2, [11,12,13,14] )
ans =
[
1
4
7
0
[
11
12
13
14
[
2
5
8
0
[
3
6
9
0
#> A.inscol(3,
ans =
[
1
[
2
[
3
]
]
]
]
[11,12])
4
5
6
11
0
0
12
0
0
7 ]
8 ]
9 ]
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Member functions
matrix
A.m // m of an m x n matrix
A.n // n of an m x n matrix
A.mn // mn of an m x n matrix
A.det // determinant
A.rank // rank of a matrix
A.inv // inverse of a matrix, A^(-1)
A.eig // eigenvalues of a matrix
and many more
#> A = [ 11,12,13; 21,22,23 ]
A =
[
11
12
13 ]
[
21
22
23 ]
#> A.m;
ans
ans
ans
A.n; A.mn;
=
=
=
#> A = [ 1,2; 4,3 ]
A =
[
1
2 ]
[
4
3 ]
#> A.det
ans =
2
3
6
#> A.inv
ans =
[
-0.6
[
0.8
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-5
0.4 ]
-0.2 ]
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■ Tuples
(a1,a2,a3,…,an)
#> (a,b) = (1,2)
a =
b =
#> (b,a) = (a,b)
b =
a =
// n-tuple
1
2
Swap values by tuples
1
2
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■ Logical (true or false), conditional
0
// false only for the zero
Non-zero // true for all non-zeros
a
a
a
a
&&
||
!&
!|
b
b
b
b
//
//
//
//
and,
or,
exclusive and,
exclusive or,
.and(a,b)
.or(a,b)
.xand(a,b)
.xor(a,b)
#> [ 0 && 0, 0 && 1, 1 && 0, 1 && 1 ]
ans = [
0
0
0
1 ]
#> [ 0 || 0, 0 || 1, 1 || 0, 1 || 1 ]
ans = [
0
1
1
1 ]
#> [ 0 !& 0, 0 !& 1, 1 !& 0, 1 !& 1 ]
ans = [
1
0
0
1 ]
#> [ 0 !| 0, 0 !| 1, 1 !| 0, 1 !| 1 ]
ans = [
0
1
1
0 ]
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■ relational
a
a
a
a
a
a
> b
< b
>= b
<= b
== b
!= b
#> 3 > 2
ans =
#> 3 < 2
ans =
#> 3 >= 2
ans =
#> 3 <= 2
ans =
#> 3 == 2
ans =
#> 3 != 2
ans =
//
//
//
//
//
//
greater than
less than
greater than or equal to
less than or equal to
equal to
not equal to
1
0
1
0
0
1
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■ relational operators for matrices only
A .mn. B
// true if A and B are of same dimension
A .eq. b
A .ne. b
// true if all a_ij == b_ij
// true if all a_ij != b_ij
A
A
A
A
.gt.
.lt.
.ge.
.le.
B
B
B
B
//
//
//
//
true
true
true
true
if
if
if
if
all
all
all
all
a_ij
a_ij
a_ij
a_ij
> b_ij
< b_ij
>= b_ij
<= b_ij
#> A = [ 1,2,3 ]; B = [ 4,5,6 ];
A = [
1
2
3 ]
B = [
4
5
6 ]
#> A .mn. B
ans =
1
#> A .mn. B'
ans =
0
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■ Array
Data can be declared as an array.
Access to its elements is done by []
double x[n] is equal to x[0],x[1],..., x[n-1]
Be careful !!! Index starts at 0
#> double x[5]
#> x[0] = 3
ans =
3
#> x[1] = 5
ans =
5
#> x
x = double [5]
[0] =
3
[1] =
5
[2] = 2.0694473e+161
[3] = 1.8875371e+219
[4] = 3.7699368e-317
Note that x[5] causes an error
since it tries to refer 6-th
element (not existing)
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■ ternary operation, a ? b : c
a ? b : c
// b if a is true, otherwise c
#> 3 ? 10 : 20;
ans =
10
#> 0 ? 10 : 20;
ans =
20
#> (a,b) = (2,3); .max(a,b); .min(a,b);
a =
2
b =
3
ans =
3
ans =
2
#> a > b ? a : b
b =
3
#> a < b ? a : b
a =
2
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■ ++, -Prefix : Variable is increased/decreased ‘before use’
++a // a = a+1
--a // a = a-1
#> a = 1; b = ++a;
a =
b =
#> a; b;
a =
b =
1
2
2
2
Postfix : Variable is increased/decreased ‘after use’
a++
// a = a+1
a-// a = a-1
#> a = 1; b = a++;
a =
b =
#> a; b;
a =
b =
1
1
2
1
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■ if-else
if(cond) stmt; // if cond == true, stmt is executed
if(cond) stmt1; // if cond == true, stmt1 is executed
else
stmt2; // if cond == false, stmt2 is executed
x = 2; s = 3;
if
(x == 1) s += 10;
else if(x == 2) s *= 10;
else if(x == 3) s /= 10;
x =
s =
s =
2nd clause is executed
2
3
30
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■ while
while(cond) stmt; // stmt while “cond” is true
i = 0;
while( ++i < 5 ) i;
i
i
i
i
i
=
=
=
=
=
0
1
2
3
4
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■ do-while
do { stmt_list } while(cond);
// run at least once
// then, stmt while “cond” is true
a = 5;
do { a = a*a; } while( a < -1 );
a =
a =
5
25
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■ for (1)
for[number] stmt; // stmt “number” times
Pibonachi sequence, x(n+2)=x(n)+x(n+1)
#> s = 0;
s
s
s
s
s
s
s
s
s
s
s
=
=
=
=
=
=
=
=
=
=
=
for[10] s += 1;
0
1
2
3
4
5
6
7
8
9
10
#> a = b = 1; for[10] { (a,b) = (b,a+b);; b; }
a
b
b
b
b
b
b
b
b
b
b
=
=
=
=
=
=
=
=
=
=
=
1
2
3
5
8
13
21
34
55
89
144
Note : double semi-colons ;; suppress
the screen output
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■ for (2)
for.n(a,b,h=1) stmt1; // stmt1 for n=a,a+h,a+2h,…
for(init; cond; expr) stmt2; // the same as C-language
#> for.j(12,3,-2) j;
#> for.k(3,7) k^2;
ans
ans
ans
ans
ans
=
=
=
=
=
j
j
j
j
j
9
16
25
36
49
=
=
=
=
=
12
10
8
6
4
#> for(j=12; j>=3; j-=2 ) j;
j
j
j
j
j
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=
=
=
=
=
12
10
8
6
4
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■ break
break;
// is used to exit from ‘while’, ‘for’, ‘switch’
i = 0;
while(1) { i; if( ++i > 5 ) break; }
i
i
i
i
i
i
i
=
=
=
=
=
=
=
0
0
1
2
3
4
5
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■ continue
continue;
// is used to move to the starting position
i = 0;
while( ++i < 5 ) { if( i == 3 ) continue; i; }
i
i
i
i
=
=
=
=
0
1
2
4
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■ goto
goto target; … target: stmt
// target must start with a string
a = 1;
a =
c =
goto skip;
b = 2;
skip: c = 3;
1
3
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■ switch-case-default
switch(n) {
case n1: stmt: break;
case n2: stmt;
default: stmt; // optional
}
#> (a,b,c) = (0,0,0)
a =
b =
c =
0
0
0
#> n = 4;
n =
4
switch(n) {
case 1: a = 10; break;
default: c = 5; // continue to the next line
case 2: b = 20; break;
case 3: c = 100;
}
c =
b =
#> a;
a
b
c
b; c;
=
=
=
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5
20
0
20
5
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■ Function (1)
double f(x) = x ;
double f(double x) = x ;
double f(double x) { return x; } // standard C grammar
#> double fn(x) = |x|*sin(x) ;
#> fn ;
double fn(x)
#> fn(3) ;
ans =
0.42336002
#> fn'(3) ; // derivative
ans =
-2.8288575
#> fn''(3) ; // 2nd-order derivative
ans =
-2.4033449
#> fn~(0,3); // integration over (0,3)
ans =
3.1110975
#> fn++( [1,2,3] ) ; // upgrade for matrix
ans = [
0.841471
1.81859
0.42336 ]
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■ Function (2) arguments
double f(a,b,c)
double f(double a,b,c)
double f(double a, double c, double c)
// C language
Repeating arguments of same types can be declared by a
leading declaration
matrix sol(a,b, vertex u,v)
has data types
matrix
matrix
matrix
vertex
vertex
sol
a
b
u
v
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■ Function (3) return type
matrix test(double n) { // matrix test(n) assumes n is also a matrix
(a,b,c) = (0,0,0);
switch(n) {
case 1: a = 10; break; // case must be followed by an integer only
default: c = 5;
case 2: b = 20; break;
case 3: c = 100;
}
return [a,b,c];
}
#> test(1);
ans = [
10
0
0 ]
#> test(2);
ans = [
0
20
0 ]
#> test(3);
ans = [
0
0
100 ]
#> test(4);
ans = [
0
20
5 ]
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■ Function (4) upgrade, matrix F(double x)
x
ut ( x)   1 
 x 2 
#> matrix ut(double x) = [ x; 1; x^2 ];
#> ut( 3 );
ans =
[
[
[
3 ]
1 ]
9 ]
#> ut++( [ 3,4,5 ] );
ans =
[
3
[
1
[
9
4
1
16
#> ut++( [ 3,4; 5,6 ] );
ans =
[
3
[
5
[
1
[
1
[
9
[
25
4
6
1
1
16
36
5 ]
1 ]
25 ]
]
]
]
]
]
]
School of Mechanical and Aerospace Engineering
Seoul National University
 3 4  


5
6

3 4   
ut ( 
)   ... 

5 6  ... 




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■ Function (5), copying arguments
Arguments are by default copied inside of a function, and thus original
variables are unchanged.
void swap1(x,y) { t = x; x = y; y = t; }
#> x = 3; y = 5;
x =
y =
3
5
#> swap1(x,y); x; y;
x =
3
y =
5
// Nothing happens to x and y.
School of Mechanical and Aerospace Engineering
Seoul National University
The Simple is the Best
www.msharpmath.com
65 / 70
■ Function (6), referring arguments
A variable can be refered by a prefix '&‘ , and the original variables are
delivered inside a function.
void swap2(&x,&y) {
t = x; x = y; y = t;
}
#> x = 3; y = 5;
x =
y =
3
5
#> swap2(x,y); x; y;
x =
5
y =
3
// Note that x and y are changed.
School of Mechanical and Aerospace Engineering
Seoul National University
The Simple is the Best
www.msharpmath.com
66 / 70
■ Function (7), array arguments
A prefix '*' is attached to array arguments and data delivery is by
reference.
void array(matrix *A)
{
A[0] = .zeros(3);
A[2] = [ 1,2,3,4; 5,6,7,8 ];
}
#> matrix An[4];
An = matrix [4]
[0] =
[
0
[
0
[
0
[1] =
[ undefined ]
[2] =
[
1
[
5
[3] =
[ undefined ]
array(An);
An;
0
0
0
0 ]
0 ]
0 ]
2
6
3
7
School of Mechanical and Aerospace Engineering
Seoul National University
4 ]
8 ]
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67 / 70
■ Function (8), recurrence
A function can call itself inside its definition, and this is the case of
recurrence function.
%> factorial function
double fac(n) {
if( n == 1 ) return 1;
return n*fac(n-1);
}
#> fac(5);
ans =
120
School of Mechanical and Aerospace Engineering
Seoul National University
The Simple is the Best
www.msharpmath.com
68 / 70
■ Function (9), dynamic default assignments
Arguments of a function can be expressed by the arguments defined
before.
double area( a, b=a ) = a*b;
// b is expressed by a
#> area ; double area( a, b=a )
#> area(3);
ans =
#> area(3,4);
ans =
9
12
School of Mechanical and Aerospace Engineering
Seoul National University
The Simple is the Best
www.msharpmath.com
69 / 70
■ Function (10), function arguments
A special grammar is innovated for a function argument as follows.
double ffx(x) = x^2;
double ggx(x) = x*sin(x);
double pftn(f(x),x) = f(x)^2;
// function argument f(x)
#> pftn(ffx,3); pftn(ggx,3);
ans =
81
ans =
0.17923371
School of Mechanical and Aerospace Engineering
Seoul National University
The Simple is the Best
www.msharpmath.com
70 / 70
Msharpmath, The Simple is the Best
Thank you !
School of Mechanical and Aerospace Engineering
Seoul National University
The Simple is the Best
www.msharpmath.com
71 / 70