MPZI_predavanje_01 -- Sage

Matematički programi za inženjere

Ishodi učenja predmeta:

nakon uspješno svladanoga predmeta moći ćete

rješavati matematičke probleme pomoću računala,
napisati jednostavnije računalne programe,
primijeniti stečena znanja u predmetima sljedećega semestra i viših godina.

Vaše obveze za ostvarivanje prava na potpis:

redovito pohađanje predavanja i vježbi,
na svakom od 2 kolokvija treba ostvariti najmanje 40 (od mogućih 100) bodova,
(može se pristupiti samo jednom popravnom kolokviju).

Ocjenjivanje:

ako na kolokvijima ostvarite

ukupno 80–119 bodova — ocjena dovoljan (2),
ukupno 120–149 bodova — ocjena dobar (3),
ukupno 150–179 bodova — ocjena vrlo dobar (4),
ukupno 180–200 bodova — ocjena odličan (5).

Sadržaj predmeta:

Predavanja:
1. Uvođenje u SageMath. Numeričko računanje
2. Izrazi, funkcije, polinomi
3. Grafika (1): Prikaz ravninskih krivulja. Prikaz prostornih krivulja. Osnovne ravninske i prostorne tvorevine
4. Grafika (2): Prikaz ploha
5. Vektori i matrice
6. Jednadžbe i nejednadžbe
7. Elementi matematičke analize (1): Granične vrijednosti funkcija. Derivacije funkcija
8. Elementi matematičke analize (2): Integrali. Uvod u diferencijalne jednadžbe
9. Uvod u programiranje
10. Nazivi, varijable, tipovi
11. Petlje (1): Petlja for
12. Petlje (2) i grananja: Ugniježđene petlje. Grananja if
13. Petlje (3): Petlja while
14. O listama i o rekurziji

Vježbe:
1. Aritmetičke operacije i matematičke funkcije
2. Izrazi, funkcije, polinomi i njihovi grafovi
3. Crtanje krivulja
4. Crtanje ploha
5. Vektori i matrice
6. Jednadžbe, sustavi jednadžbi i nejednadžbe
7. Limesi i derivacije
8. Integrali i diferencijalne jednadžbe
9. Prvi kolokvij
10. Prvi programi
11. Petlja for
12. Naredba if
13. Dvostruka petlja for
14. Petlja while
15. Drugi kolokvij

Literatura:

Sage PREP Tutorials
M. O'Sullivan & D. Monarres: Sage Tutorial

Dopunska literatura:

Sage Reference Manual
G. Bard: Sage for Undergraduates (.pdf.zip)

Uvođenje u SageMath

SageMath (Sage) — otvoreni programski sustav za simboličku matematiku (Computer Algebra System, CAS)

(Drugi simbolički programski sustavi: Macsyma, Reduce, Mathematica, Maxima, MATLAB, SymPy, ...)

Osnovne mogućnosti:

numeričko računanje,
simboličko baratanje matematičkim izrazima,
grafički prikaz funkcija i podataka,
programiranje.

Numeričko računanje (number crunching — drobljenje brojeva):

2^999 + 1

53575430359313366047421252453000090528070240585276680372187519418517552556246806\
12465991894078479290637973364587765734125935726428461570217992288787349287401967\
28388741211549271053730253118557093897709107652323749179097063369938377958277197\
3038531457285598238843271083830214915826312193418602834034689

5357543035931336604742125245300009052807024058527668037218751941851755255624680612465991894078479290637973364587765734125935726428461570217992288787349287401967283887412115492710537302531185570938977091076523237491790970633699383779582771973038531457285598238843271083830214915826312193418602834034689

nelinearne jednadžbe:

npr. kvadratna jednadžba $x^2 + \frac{x}{2} - 1 = 0$

plot (x^2 + x/2. - 1., -2., 2.)

find_root (x^2 + x/2. - 1. == 0., 0., 2.)

0.7807764064044151

0.7807764064044151

find_root (x^2 + x/2. - 1. == 0., -2., 0.)

-1.2807764064044151

-1.2807764064044151

matrice, vektori, sustavi linearnih jednadžbi:

A = matrix ([[4., 1., 1., 1.], [1., 4., 1., 1.], [1., 1., 4., 1.], [1., 1., 1., 4.]])

A

[4.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000]
[1.00000000000000 4.00000000000000 1.00000000000000 1.00000000000000]
[1.00000000000000 1.00000000000000 4.00000000000000 1.00000000000000]
[1.00000000000000 1.00000000000000 1.00000000000000 4.00000000000000]

[4.00000000000000 1.00000000000000 1.00000000000000 1.00000000000000]
[1.00000000000000 4.00000000000000 1.00000000000000 1.00000000000000]
[1.00000000000000 1.00000000000000 4.00000000000000 1.00000000000000]
[1.00000000000000 1.00000000000000 1.00000000000000 4.00000000000000]

show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 4.00000000000000 & 1.00000000000000 & 1.00000000000000 & 1.00000000000000 \\ 1.00000000000000 & 4.00000000000000 & 1.00000000000000 & 1.00000000000000 \\ 1.00000000000000 & 1.00000000000000 & 4.00000000000000 & 1.00000000000000 \\ 1.00000000000000 & 1.00000000000000 & 1.00000000000000 & 4.00000000000000 \end{array}\right)$

b = vector ([-3., 3., 3., -3.])
b

(-3.00000000000000, 3.00000000000000, 3.00000000000000, -3.00000000000000)

(-3.00000000000000, 3.00000000000000, 3.00000000000000, -3.00000000000000)

matricom $\mathbf{A}$ i vektorom $\mathbf{b}$ definiran je sustav linearnih jednadžbi $\mathbf{A}\cdot \mathbf{x} = \mathbf{b}$ :

$4\,x_0 + x_1 + x_2 + x_3 \,=\, -3$

$x_0 + 4\,x_1 + x_2 + x_3 \,=\, 3$

$x_0 + x_1 + 4\,x_2 + x_3 \,=\, 3$

$x_0 + x_1 + x_2 + 4\,x_3 \,=\, -3$

x = A.solve_right (b)
x

(-1.00000000000000, 1.00000000000000, 1.00000000000000, -1.00000000000000)

(-1.00000000000000, 1.00000000000000, 1.00000000000000, -1.00000000000000)

x = A \ b
x

(-1.00000000000000, 1.00000000000000, 1.00000000000000, -1.00000000000000)

(-1.00000000000000, 1.00000000000000, 1.00000000000000, -1.00000000000000)

A * x

(-3.00000000000000, 3.00000000000000, 3.00000000000000, -3.00000000000000)

(-3.00000000000000, 3.00000000000000, 3.00000000000000, -3.00000000000000)

A * x - b

(0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000)

(0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000)

det (A)

189.000000000000

189.000000000000

var ('x')

numerička integracija:

geometrijsko značenje određenoga integrala — ploština površine „ispod” krivulje

plot (sin (x)^5 + sin(x)^3 + sin(x), 0, pi, fill = True)

ni = numerical_integral (sin (x)^5 + sin(x)^3 + sin(x), 0, pi)
ni

(4.3999999999999995, 4.884981308350688e-14)

(4.3999999999999995, 4.884981308350688e-14)

ni[0] - ni[1], ni[0] + ni[1],

(4.399999999999951, 4.400000000000048)

(4.399999999999951, 4.400000000000048)

Simboličko baratanje matematičkim izrazima:

var ('y z a b c')

(y, z, a, b, c)

(y, z, a, b, c)

„pojednostavnjenje” izraza:

show ((x - y + z)^2)

$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x - y + z\right)}^{2}$

expand ((x - y + z)^2)

x^2 - 2*x*y + y^2 + 2*x*z - 2*y*z + z^2

x^2 - 2*x*y + y^2 + 2*x*z - 2*y*z + z^2

show (_)

$\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} - 2 \, x y + y^{2} + 2 \, x z - 2 \, y z + z^{2}$

factor (x^2 - 2*x*y + 2*x*z + y^2 - 2*y*z + z^2)

(x - y + z)^2

(x - y + z)^2

show (_)

$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x - y + z\right)}^{2}$

„točno” rješavanje jednadžbi:

xx = solve (x^2 + x/2. - 1. == 0., x)
xx

[x == -1/4*sqrt(17) - 1/4, x == 1/4*sqrt(17) - 1/4]

[x == -1/4*sqrt(17) - 1/4, x == 1/4*sqrt(17) - 1/4]

show (xx)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = -\frac{1}{4} \, \sqrt{17} - \frac{1}{4}, x = \frac{1}{4} \, \sqrt{17} - \frac{1}{4}\right]$

N (xx[0].rhs()); N (xx[1].rhs())

-1.28077640640442
0.780776406404415

-1.28077640640442
0.780776406404415

N (xx[0].rhs(), digits = 50); N (xx[1].rhs(), digits = 50)

-1.2807764064044151374553524639935192562867998063434
0.78077640640441513745535246399351925628679980634341

-1.2807764064044151374553524639935192562867998063434
0.78077640640441513745535246399351925628679980634341

ranije dobivena numerička rješenja: -1,2807764064044151 i 0,7807764064049022

simboličko rješavanje jednadžbi:

$a x^2 + b x + c = 0$

solve (a*x^2 + b*x + c == 0, x)

[x == -1/2*(b + sqrt(b^2 - 4*a*c))/a, x == -1/2*(b - sqrt(b^2 - 4*a*c))/a]

[x == -1/2*(b + sqrt(b^2 - 4*a*c))/a, x == -1/2*(b - sqrt(b^2 - 4*a*c))/a]

show (_)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = -\frac{b + \sqrt{b^{2} - 4 \, a c}}{2 \, a}, x = -\frac{b - \sqrt{b^{2} - 4 \, a c}}{2 \, a}\right]$

definiranje funkcija:

$f(x) = -x^3 - 3 x^2 + 1$

f(x) = -x^3 - 3 * x^2 + 1

show (f)

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -x^{3} - 3 \, x^{2} + 1$

f(-0.4), f(0.8)

(0.584000000000000, -1.43200000000000)

(0.584000000000000, -1.43200000000000)

show (f(a))

$\newcommand{\Bold}[1]{\mathbf{#1}}-a^{3} - 3 \, a^{2} + 1$

deriviranje:

diff (cos (x), x)

-sin(x)

-sin(x)

df(x) = diff (f(x), x)
show (df)

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -3 \, x^{2} - 6 \, x$

df(-0.4), df(0.8)

(1.92000000000000, -6.72000000000000)

(1.92000000000000, -6.72000000000000)

geometrijsko značenje derivacije — brzina rasta ili pada krivulje koja se mjeri nagibima njezinih tangenata

p1 = plot (f, -1.5, 1.2)
p2 = plot (df, -1.5, 1.2, color = 'red')
p1 + p2

integriranje:

neodređeni integral:

integral (-sin (x), x)

cos(x)

cos(x)

integral (df(x), x)

-x^3 - 3*x^2

-x^3 - 3*x^2

show (_)

$\newcommand{\Bold}[1]{\mathbf{#1}}-x^{3} - 3 \, x^{2}$

show (f(x))

$\newcommand{\Bold}[1]{\mathbf{#1}}-x^{3} - 3 \, x^{2} + 1$

integral (sin (x)^5 + sin(x)^3 + sin(x), x)

-1/5*cos(x)^5 + cos(x)^3 - 3*cos(x)

-1/5*cos(x)^5 + cos(x)^3 - 3*cos(x)

show (_)

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{5} \, \cos\left(x\right)^{5} + \cos\left(x\right)^{3} - 3 \, \cos\left(x\right)$

određeni integral:

integral (sin (x)^5 + sin(x)^3 + sin(x), x, 0, pi)

22/5

22/5

N (_)

4.40000000000000

4.40000000000000

numeričkom je integracijom dobivena vrijednost 4,3999999999999995 (točnije, dobiveno je da je vrijednost u intervalu [4,399999999999951; 4,400000000000048])

Grafički prikaz funkcija i podataka:

plot3d (log (cos (x) / cos (y)), (x, -1, 1), (y, -1, 1))

Click on 3-D image to make it live. Right-click on live image for a control menu.

Programiranje:

[i^2 for i in srange (10)]

[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

def hypar (x, y) : 
    return x * y
    
A = matrix (RR, 9, 9)
h = 2./8

for i in srange (9) :
    for j in srange (9) :
        A[i,j] = hypar (-1. + i*h, -1. + j*h)

matrix_plot (A)

list_plot3d (A, mesh = True).rotateZ (-pi/5)

Numeričko računanje

Aritmetičke operacije:

zbrajanje: +
oduzimanje: –
množenje: *
dijeljenje: /
- cjelobrojno: //, ostatak: %
potenciranje: ^, **

operacije imaju „klasični” prioritet:

2 + 3 * 4 - 15

-1

-1

funkcijski poziv ima viši prioritet:

sin (pi/2)^2 + cos (pi/2)^2

promjena prioriteta:

(2 + 3) * (4 - 15)

-55

-55

2^4/2

2^(4/2)

treba upotrijebiti okrugle zagrade, (); uglate i vitičaste zagrade, [] i {}, imaju druga značenja:

(2 + 3) * [4 - 15]

[-11, -11, -11, -11, -11]

[-11, -11, -11, -11, -11]

[2 + 3] * (4 - 15)

[]

[]

[2 + 3] * [4 - 15]

Traceback (click to the left of this block for traceback)
...
TypeError: can't multiply sequence by non-int of type 'list'

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_63.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("WzIgKyAzXSAqIFs0IC0gMTVd"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpmlyAJN/___code___.py", line 3, in <module>
    exec compile(u'[_sage_const_2  + _sage_const_3 ] * [_sage_const_4  - _sage_const_15 ]
  File "", line 1, in <module>
    
TypeError: can't multiply sequence by non-int of type 'list'

cijeli, racionalni i „realni” brojevi:

12/2

13/2

13/2

13/2

13//2

13%2

(13//2)*2 + (13%2)

13./2.

6.50000000000000

6.50000000000000

13./2

6.50000000000000

6.50000000000000

13/2.

6.50000000000000

6.50000000000000

N (13/2)

6.50000000000000

6.50000000000000

N (2/13)

0.153846153846154

0.153846153846154

N (2/13, digits = 50)

0.15384615384615384615384615384615384615384615384615

0.15384615384615384615384615384615384615384615384615

N (13/2, digits = 50)

6.5000000000000000000000000000000000000000000000000

6.5000000000000000000000000000000000000000000000000

ako se broj znamenaka (digits) ne navede, funkcija N „realni” broj ispisuje s petnaest značajnih znamenaka

13^(-13)

1/302875106592253

1/302875106592253

13.^(-13)

3.30169095523012e-15

3.30169095523012e-15

u računanju s „realnim” brojevima potreban je oprez:

0.1 * 10

1.00000000000000

1.00000000000000

(0.1 * 10) == 1.

True

True

0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1

1.00000000000000

1.00000000000000

(0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1) == 1.

False

False

print (0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1).str (truncate = False)

0.99999999999999989

0.99999999999999989

Relacijske operacije:

jednako: ==
nije jednako: !=
veće: >
veće ili jednako: >=
manje: <
manje ili jednako: <=

2 > 3

False

False

2 <= 3

True

True

2 < 2

False

False

2 <= 2

True

True

prioritet relacijskih operatora niži je od prioriteta aritmetičkih:

2^3 == 2^4 / 2

True

True

2^4 / 2 == 2^(4/2)

False

False

2 + 3 * 4 - 15 != (2 + 3) * (4 - 15)

True

True

0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 == 1.

False

False

Logičke operacije:

i: and
ili: or

(-1 < 2) and (2 < 3)

True

True

prioritet logičkih operatora niži je od prioriteta relacijskih:

-1 > 2 and 2 < 3

False

False

-1 < 2 and 2 > 3

False

False

-1 > 2 and 2 > 3

False

False

MPZI_predavanje_01

2492 days ago by fresl