PPPK_03 -- Sage

Algebarske strukture

Monoid: $\mathcal{M} = (\mathcal{S}, *)$, pri čemu su $\mathcal{S}$ skup i $*$ binarna operacija

zatvorenost: $x, y \in \mathcal{S} \; \Rightarrow \; x*y \in \mathcal{S}$
asocijativnost: $(x * y) * z = x * (y * z)$
neutralni (jedinični) element: $\exists e \in \mathcal{S} \; : \; x * e = e * x = x \;\;\: \forall x \in\mathcal{S}$

Grupa: $\mathcal{G} = (\mathcal{S}, *)$

monoid
inverzni element: $\forall x \in \mathcal{S}\;\;\; \exists y \in \mathcal{A} \; : \; x * y = y * x = e$; česte oznake: $y = x^{-1}$ ili $y = -x$

Abelova ili komutativna grupa

grupa
komutativnost: $x * y = y * x$

Primjeri

skupovi $\mathbb{Z},\; \mathbb{Q}, \; \mathbb{R}$ i operacija $+$ : Abelove grupe
skup $\mathbb{N}_0 = \mathbb{N} \cup \{0\}$ i operacija $+$ : monoid u kojem vrijedi komutativnost (nije grupa jer nema inverznog elementa)
skupovi $\mathbb{Q} \setminus \{0\},\; \mathbb{R}\setminus \{0\}, \; \mathbb{Q}^+, \; \mathbb{R}^+$ i operacija $\cdot$ : Abelove grupe
skup regularnih matrica tipa $n\times n$ i operacija množenja : grupa (ali ne Abelova, jer ${\mathbf{AB}} \ne {\mathbf{BA}}$)
skup bijekcija na skupu $\mathcal{S}$ i kompozicija funkcija : grupa (ali ne Abelova, jer $f\circ g \ne g \circ f$ za $f, g : \mathcal{S} \to \mathcal{}S$)

Prsten: $\mathcal{R} = (\mathcal{S}, +, \cdot)$, pri čemu su $+$ i $\cdot$ binarne operacije

zatvorenost obje operacije
$(\mathcal{S}, +)$ Abelova grupa
asocijativnost operacije $\cdot$
distributivnost: $x\cdot (y + z) = x\cdot y + x \cdot z$ i $(x + y)\cdot z = x\cdot z + y \cdot z$

Komutativni prsten

prsten
komutativnost operacije $\cdot$

Prsten s jedinicom

prsten
neutralni element za $\cdot$ : $x\cdot 1 = 1\cdot x = x$; $1 \ne 0$, gdje je $0$ neutralni element za $+$

Prsten s dijeljenjem

prsten s jedinicom
inverzni element za $\cdot$ : $\forall x \ne 0\;\;\; \exists y \; : \; x \cdot y = y \cdot x = 1$

Polje — komutativni prsten s dijeljenjem

Primjeri

$(\mathbb{N}_0, +, \cdot)$ komutativni poluprsten s jedinicom (poluprsten ima svojstva prstena, osim što je 2. slabije: $(\mathcal{S}, +)$ komutativni monoid)
$(\mathbb{Z}, +, \cdot)$ komutativni prsten s jedinicom
$(\mathbb{Q}, +, \cdot)$, $(\mathbb{R}, +, \cdot)$, $(\mathbb{C}, +, \cdot)$ polja
regularne matrice tipa $n\times n$ s operacijama zbrajanja i množenja : prsten s dijeljenjem (množenje nije komutativno)

Vektorski prostor $\boldsymbol{\mathcal{V}}$ nad poljem $\boldsymbol{\mathcal{S}}$: $(\mathcal{V},\, \mathcal{S},\, +,\, \cdot)$, pri čemu su

$\mathcal{V}$ - skup vektora,
$\mathcal{S}$ - polje skalara,
$+$ - zbrajanje vektora $+ : \mathcal{V}\times \mathcal{V} \to \mathcal{V}$,
$\cdot$ - množenje skalarom $\cdot : \mathcal{S}\times \mathcal{V} \to \mathcal{V}$

zbrajanje je Abelova grupa (neutralni element $\mathbf{0}$, inverzni element $-\mathbf{x}$)
zatvorenost množenja skalarom: $\alpha \in \mathcal{S},\: \mathbf{x}\in\mathcal{V} \; \Rightarrow \; \alpha\,\mathbf{x} = \alpha\cdot\mathbf{x}\in \mathcal{V}$
$(\alpha\,\beta)\,\mathbf{x} = \alpha\,(\beta\,\mathbf{x}) \;\;\;\: \forall \alpha,\, \beta \in \mathcal{S}\;\;\; \text{i} \;\;\; \forall \mathbf{x}\in\mathcal{V}$
$\alpha\,(\mathbf{x} + \mathbf{y}) = \alpha\,\mathbf{x} + \alpha\mathbf{y} \;\;\;\: \forall \alpha \in \mathcal{S}\;\;\; \text{i} \;\;\; \forall \mathbf{x},\,\mathbf{y}\in\mathcal{V}$
$(\alpha + \beta)\,\mathbf{x} = \alpha\,\mathbf{x} + \beta\,\mathbf{x} \;\;\;\: \forall \alpha,\, \beta \in \mathcal{S}\;\;\; \text{i} \;\;\; \forall \mathbf{x}\in\mathcal{V}$
$\exists\, 1\! \in \mathcal{S} \; : \; 1\,\mathbf{x} = \mathbf{x} \;\;\; \forall \mathbf{x}\in\mathcal{V}$

Primjeri

„klasični” vektori u ravnini i prostoru (nad poljem $\mathbb{R}$)
poopćenje „klasičnih” vektora u $n$-dimenzionalnom prostoru $\mathbb{R}^n$
matrice tipa $m\times n$ (nad poljem $\mathbb{R}$)
skup svih funkcija $f : [0,1] \to \mathbb{R}$, pri čemu su $(f+g)(x) = f(x) + g(x)$ i $(\alpha f)(x) = \alpha\, f(x)$ za $\alpha \in \mathbb{R}$
skup svih neprekinutih funkcija $f : [0,1] \to \mathbb{R}$
skup svih funkcija $f : [0,1] \to \mathbb{R}$ s neprekinutom prvom derivacijom
skup svih polinoma s realnim koeficijentima

Slobodni modul: $(\mathcal{V},\, \mathcal{S},\, +,\, \cdot)$, ali je, za razliku od vektorskog prostora, $\mathcal{S}$ prsten s jedinicom

Algebra (nad poljem): vektorski prostor s množenjem

Primjeri

kompleksni brojevi — komutativna asocijativna algebra
matrice tipa $n\times n$ — nekomutativna asocijativna algebra
prostor $\mathbb{R}^3$ s vektorskim produktom — nekomutativna neasocijativna algebra

1

parent (1)

Integer Ring

Integer Ring

1.parent()

Integer Ring

Integer Ring

1/2

1/2

1/2

show (1/2)

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2}$

parent (1/2)

Rational Field

Rational Field

1.0

1.00000000000000

1.00000000000000

parent (1.)

Real Field with 53 bits of precision

Real Field with 53 bits of precision

RealField (80)

Real Field with 80 bits of precision

Real Field with 80 bits of precision

a = RealField (80) (1.)
a

1.0000000000000000000000

1.0000000000000000000000

parent (a)

Real Field with 80 bits of precision

Real Field with 80 bits of precision

RR

Real Field with 53 bits of precision

Real Field with 53 bits of precision

CC

Complex Field with 53 bits of precision

Complex Field with 53 bits of precision

Oznake skupova brojeva:

ZZ — $\mathbb{Z}$, prsten cijelih brojeva (Integer Ring)
QQ — $\mathbb{Q}$, polje racionalnih brojeva (Rational Field)
RR — $\mathbb{R}$, polje realnih brojeva (Real Field) s točnošću zapisa od 53 bita (približno 16 značajnih dekadskih znamenaka)
RealField (n) — $\mathbb{R}$, polje realnih brojeva (Real Field) s točnošću zapisa od $n$ bitova (približno $n\cdot \log_{10}2$ značajnih dekadskih znamenaka)
CC — $\mathbb{C}$, polje kompleksnih brojeva (Complex Field) s točnošću zapisa od 53 bita za realni i za imaginarni dio

v = zero_vector (RR, 5)
v

(0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000,
0.000000000000000)

(0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000)

parent (v)

Vector space of dimension 5 over Real Field with 53 bits of precision

Vector space of dimension 5 over Real Field with 53 bits of precision

v2 = zero_vector (RealField (80), 5)
v2

(0.00000000000000000000000, 0.00000000000000000000000,
0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000)

(0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000)

parent (v2)

Vector space of dimension 5 over Real Field with 80 bits of precision

Vector space of dimension 5 over Real Field with 80 bits of precision

v3 = vector (RealField (14), 5)
v3

(0.000, 0.000, 0.000, 0.000, 0.000)

(0.000, 0.000, 0.000, 0.000, 0.000)

v3.parent()

Vector space of dimension 5 over Real Field with 14 bits of precision

Vector space of dimension 5 over Real Field with 14 bits of precision

v4 = vector ([1/2, 2/3, 3/4])
v4

(1/2, 2/3, 3/4)

(1/2, 2/3, 3/4)

show (v4)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{1}{2},\,\frac{2}{3},\,\frac{3}{4}\right)$

parent (v4)

Vector space of dimension 3 over Rational Field

Vector space of dimension 3 over Rational Field

parent (vector ([1/2, 0.5]))

Vector space of dimension 2 over Real Field with 53 bits of precision

Vector space of dimension 2 over Real Field with 53 bits of precision

vector ([1/2, 0.5])

(0.500000000000000, 0.500000000000000)

(0.500000000000000, 0.500000000000000)

v5 = vector ([13, 14, 15])
print v5
parent (v5)

(13, 14, 15)
Ambient free module of rank 3 over the principal ideal domain Integer Ring

(13, 14, 15)
Ambient free module of rank 3 over the principal ideal domain Integer Ring

v6 = free_module_element ([13, 17, 23])
parent (v6)

Ambient free module of rank 3 over the principal ideal domain Integer Ring

Ambient free module of rank 3 over the principal ideal domain Integer Ring

m = identity_matrix (RR, 3)
m

[ 1.00000000000000 0.000000000000000 0.000000000000000]
[0.000000000000000  1.00000000000000 0.000000000000000]
[0.000000000000000 0.000000000000000  1.00000000000000]

[ 1.00000000000000 0.000000000000000 0.000000000000000]
[0.000000000000000  1.00000000000000 0.000000000000000]
[0.000000000000000 0.000000000000000  1.00000000000000]

parent (m)

Full MatrixSpace of 3 by 3 dense matrices over Real Field with 53 bits of
precision

Full MatrixSpace of 3 by 3 dense matrices over Real Field with 53 bits of precision

m2 = random_matrix (RealField (80), 4, 3)
show (m2)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 0.94650602594839243598548 & -0.70878936039744677017903 & 0.60456103424453900678187 \\ 0.14120833919422517495153 & -0.43219527995240299331888 & -0.70483608948932100297957 \\ 0.34247803404593368570788 & 0.68445199607452818453011 & 0.80320704722800120674245 \\ 0.42730905416264094240416 & -0.99890640461970851049134 & 0.90340163985851571211700 \end{array}\right)$

parent (m2)

Full MatrixSpace of 4 by 3 dense matrices over Real Field with 80 bits of
precision

Full MatrixSpace of 4 by 3 dense matrices over Real Field with 80 bits of precision

Cijeli brojevi

-2 + 3*4

12/2

13/2

13/2

13/2

13//2

13 % 2

q, r = 13.quo_rem (2)
q, r

(6, 1)

(6, 1)

q*2 + r

N (13/2)

6.50000000000000

6.50000000000000

parent (_)

Real Field with 53 bits of precision

Real Field with 53 bits of precision

N (13/2, digits = 2)

6.5

6.5

parent (_)

Real Field with 10 bits of precision

Real Field with 10 bits of precision

13^13

302875106592253

302875106592253

13^(-13)

1/302875106592253

1/302875106592253

N (13^(-13))

3.30169095523012e-15

3.30169095523012e-15

71.str (base = 2)

'1000111'

'1000111'

binarni prikaz:

$(1000111)_2 = 1\cdot 2^0 + 1\cdot 2^1 + 1\cdot 2^2 + 0\cdot 2^3 + 0\cdot 2^4 + 0\cdot 2^5 + 1\cdot 2^6 = 1 + 2 + 4 + 0\cdot 8 + 0\cdot 16 + 0\cdot 32 + 64 = 71$

(-71).str (base = 2)

'-1000111'

'-1000111'

(2^32 - 71).str (base = 2)

'11111111111111111111111110111001'

'11111111111111111111111110111001'

dvojni komplement:

negativni broj $-x$: $2^{32} - x$

$\phantom{+}(\phantom{1}00000000000000000000000001000111)_2$

$+(\phantom{1}11111111111111111111111110111001)_2$

$=\!(100000000000000000000000000000000)_2$

$x \,-\, y \:=\: x \,+\, (-y)$: $x \,+\, (2^{32} - y) \:=\: 2^{32} \,-\, (y - x)$

„Realni” brojevi

I am HAL Nine Thousand computer Production Number 3. I became operational at the Hal Plant in Urbana, Illinois, on January 12, 1997.

The quick brown fox jumps over the lazy dog.

The rain in Spain is mainly in the plain.

Dave—are you still here?

Did you know that the square root of 10 is 3.162277660168379?

Log 10 to the base $e$ is 0.43429481903252$\,\ldots$

correction, that is log $e$ to the base 10$\,\ldots$

The reciprocal of 3 is 0.3333333333333333333$\,\ldots$

2 times 2 is$\:\ldots\:$2 times 2 is$\,\ldots$

approximately 4.10101010101010$\,\ldots$

I seem to be having difficulty$\,\ldots$

—HAL, in 2001: A Space Odyssey

13./2.

6.50000000000000

6.50000000000000

13./2

6.50000000000000

6.50000000000000

13/2.

6.50000000000000

6.50000000000000

„Realne” se konstante (s točnošću zapisa od 53 bita) mogu pisati na pet načina:

kao decimalni brojevi s točkom (može se izostaviti dio ispred ili iza točke, ali točka mora postojati, jer je inače riječ o cijelom broju):

3.14, -1.41, 0., .51

(3.14000000000000, -1.41000000000000, 0.000000000000000, 0.510000000000000)

(3.14000000000000, -1.41000000000000, 0.000000000000000, 0.510000000000000)

u eksponencijalnom obliku:

1.3e13, 13E-14

(1.30000000000000e13, 1.30000000000000e-13)

(1.30000000000000e13, 1.30000000000000e-13)

pomoću funkcije RR():

RR (1), RR (1/2)

(1.00000000000000, 0.500000000000000)

(1.00000000000000, 0.500000000000000)

pomoću funkcija N(), n(), numerical_approx():

N (1/2), n (1/3), (1/4).n(), numerical_approx (1/11), (1/13).numerical_approx()

(0.500000000000000,
 0.333333333333333,
 0.250000000000000,
 0.0909090909090909,
 0.0769230769230769)

(0.500000000000000,
 0.333333333333333,
 0.250000000000000,
 0.0909090909090909,
 0.0769230769230769)

RDF (-1), RDF (-1/3), RDF (-1/4)

(-1.0, -0.3333333333333333, -0.25)

(-1.0, -0.3333333333333333, -0.25)

Prva četiri načina daju brojeve koji pripadaju tipu RealField(53), a peti broj koji pripada tipu RealDoubleField:

parent (1.); parent (1E1); parent (RR (1)); parent (n (1)); parent (RDF (1))

Real Field with 53 bits of precision
Real Field with 53 bits of precision
Real Field with 53 bits of precision
Real Field with 53 bits of precision
Real Double Field

Real Field with 53 bits of precision
Real Field with 53 bits of precision
Real Field with 53 bits of precision
Real Field with 53 bits of precision
Real Double Field

Pomoću funkcija RealField()() i N() mogu se zadati „realni” brojevi drugih točnosti:

RealField(13) (-6/5)

-1.20

-1.20

RF170 = RealField(170)
RF170

Real Field with 170 bits of precision

Real Field with 170 bits of precision

RF170 (-6/5)

-1.2000000000000000000000000000000000000000000000000

-1.2000000000000000000000000000000000000000000000000

N (-6/5, 170)

-1.2000000000000000000000000000000000000000000000000

-1.2000000000000000000000000000000000000000000000000

N (-6/5, digits = 50)

-1.2000000000000000000000000000000000000000000000000

-1.2000000000000000000000000000000000000000000000000

Realni brojevi (u matematici):

neprebrojivi (kardinalni broj skupa $\mathbb{R}$: $\mathsf{C} = 2^{\aleph_0}\!$),
racionalni i iracionalni ($\sqrt{2},\; \pi,\; e$)

Racionalni brojevi:

prebrojivi (kardinalni broj skupova $\mathbb{N}, \mathbb{Z}$ i $\mathbb{Q}$: $\aleph_0\!$),
mogu se napisati u obliku $\dfrac{m}{n}$, gdje su $m \in \mathbb{Z}$ i $n \in \mathbb{N}$,
ovisno o bazi, u zapisu sa zarezom (ili, u engleskom, s točkom) mogu imati konačan ili beskonačan broj znamenaka; ako je znamenaka beskonačno mnogo, one se prije ili kasnije počnu ponavljati: $\dfrac{1}{3} = 0,\!333\ldots = 0,\!\dot{3}$; $\dfrac{1}{10} = 0,\!1 = (0,\!00011001100110011\ldots)_2 = (0,\!0\overline{0011})_2$

13/7

13/7

13/7

RR (13/7)

1.85714285714286

1.85714285714286

RR (130/7)

18.5714285714286

18.5714285714286

RR (130000/7)

18571.4285714286

18571.4285714286

RR (13/700)

0.0185714285714286

0.0185714285714286

RR (13/700000)

0.0000185714285714286

0.0000185714285714286

RR (13/700000000000000000000000000000000000)

1.85714285714286e-35

1.85714285714286e-35

Značajnim znamenkama broja $x\ne 0$ u zapisu sa zarezom nazivaju se prva znamenka slijeva različita od nule i sve znamenke desno od nje do kraja zapisa; za $x = 0$ sve su znamenke u zapisu značajne.

RR (0)

0.000000000000000

0.000000000000000

Prikaz s pomičnim (plivajućim) zarezom (u engleskom: s plivajućom točkom — floating point)

dekadski sustav: $x = \pm m \cdot 10^e$

$m$ — normalizirana mantisa, $1 \le m < 10$ ili $m = 0$

$e$ — eksponent

$123456780000000000000,\!0 = 1,\!2345678 \cdot 10^{20}$

$123,\!45678 = 1,\!2345678 \cdot 10^2$

$0,\!00012345678 = 1,\!2345678 \cdot 10^{-4}$

$0,\!000000000000000000012345678 = 1,\!2345678 \cdot 10^{-20}$

normalizacija omogućava zapis najvećeg broja značajnih znamenaka ako je broj znamenaka u prikazu broja ograničen

binarni sustav (najjednostavnije): $x = s \cdot m \cdot 2^e$

$s \in \{ -1,\, 1 \}$

$1 \le m < 2$ ili $m = 0$

ako je $m \in [1,\, 2)$, prva je znamenka u binarnom prikazu uvijek 1 $\rightarrow$ skriveni bit

binarni sustav (u GNU MPFR-u i SageMath-u (RealField())): $x = s \cdot m \cdot 2^{e - p}$

$p$ — točnost (broj bitova mantise)

$2^{p-1} \le m < 2^p$ ili $m = 0$

$-2^B + 1 \: \le \: e \: \le \: 2^B - 1$

($B = 30$ na 32-bitnim računalima, $B = 62$ na 64-bitnim računalima)

[GNU MPFR — GNU Multiple Precision Floating-Point Reliable Library; GNU (GNU's Not Unix!) — an extensive collection of free software]

-1.

-1.00000000000000

-1.00000000000000

sign, mantissa, e_minus_p = (-1.).sign_mantissa_exponent()
sign, mantissa, e_minus_p, 2^e_minus_p

(-1, 4503599627370496, -52, 1/4503599627370496)

(-1, 4503599627370496, -52, 1/4503599627370496)

sign * mantissa * 2^e_minus_p

-1

-1

1./3

0.333333333333333

0.333333333333333

sign, mantissa, e_minus_p = (1./3).sign_mantissa_exponent()
sign, mantissa, e_minus_p, 2^e_minus_p

(1, 6004799503160661, -54, 1/18014398509481984)

(1, 6004799503160661, -54, 1/18014398509481984)

show (sign * mantissa * 2^e_minus_p)

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{6004799503160661}{18014398509481984}$

show ((1./3).exact_rational())

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{6004799503160661}{18014398509481984}$

1./10

0.100000000000000

0.100000000000000

sign, mantissa, e_minus_p = (1./10).sign_mantissa_exponent()
sign, mantissa, e_minus_p, 2^e_minus_p

(1, 7205759403792794, -56, 1/72057594037927936)

(1, 7205759403792794, -56, 1/72057594037927936)

show (sign * mantissa * 2^e_minus_p)

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3602879701896397}{36028797018963968}$

show ((1./10).exact_rational())

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3602879701896397}{36028797018963968}$

a = RealField (80) (-1.)
a

-1.0000000000000000000000

-1.0000000000000000000000

parent (a)

Real Field with 80 bits of precision

Real Field with 80 bits of precision

sign, mantissa, e_minus_p = a.sign_mantissa_exponent()
sign, mantissa, e_minus_p, 2^e_minus_p

(-1, 604462909807314587353088, -79, 1/604462909807314587353088)

(-1, 604462909807314587353088, -79, 1/604462909807314587353088)

sign * mantissa * 2^e_minus_p

-1

-1

a/3

-0.33333333333333333333333

-0.33333333333333333333333

sign, mantissa, e_minus_p = (a/3).sign_mantissa_exponent()
sign, mantissa, e_minus_p, 2^e_minus_p

(-1, 805950546409752783137451, -81, 1/2417851639229258349412352)

(-1, 805950546409752783137451, -81, 1/2417851639229258349412352)

show (sign * mantissa * 2^e_minus_p)

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{805950546409752783137451}{2417851639229258349412352}$

show ((a/3).exact_rational())

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{805950546409752783137451}{2417851639229258349412352}$

a/3, parent (a/3)

(-0.33333333333333333333333, Real Field with 80 bits of precision)

(-0.33333333333333333333333, Real Field with 80 bits of precision)

a/3., parent (a/3.)

(-0.333333333333333, Real Field with 53 bits of precision)

(-0.333333333333333, Real Field with 53 bits of precision)

točnost (precision) prikaza s pomičnim zarezom ovisi o broju bitova mantise (binarnih značajnih znamenaka)

veza između broja značajnih znamenaka u binarnom i u dekadskom zapisu:

ako je u binarnom sustavu mantisa broja zapisana s $b$ bitova (binarnih znamenaka), onda će u dekadskom sustavu broj imati približno $b\cdot \log_{10}2$ značajnih znamenaka
za zapis $d$ dekadskih značajnih znamenaka potrebno je približno $d\cdot \log_{2}10$ bitova

s2 = 0
for i in xrange (53) : 
    a = 2^i
    s2 += a
s2; 2^53

9007199254740991
9007199254740992

9007199254740991
9007199254740992

s10 = 0
for i in xrange (16) : 
    a = 9 * 10^i
    s10 += a
s10; 10^16

9999999999999999
10000000000000000

9999999999999999
10000000000000000

2^53; 10^16

9007199254740992
10000000000000000

9007199254740992
10000000000000000

53 * log (2., 10)

15.9545897701910

15.9545897701910

16 * log (10., 2)

53.1508495181978

53.1508495181978

def sigdigs (s, a, b) :
    return round (s * log (a, b))

def sigdigs_dec2bin (d) :
    return sigdigs (d, 10., 2.)

def sigdigs_bin2dec (b) :
    return sigdigs (b, 2., 10.)

sigdigs_bin2dec (53)

sigdigs_dec2bin (16)

sigdigs_bin2dec (80)

sigdigs_dec2bin (24)

SageMath je pri ispisu brojeva na „(ne)sigurnoj strani”:

parent (4./3)

Real Field with 53 bits of precision

Real Field with 53 bits of precision

print (4./3).str (truncate = False) 
print (4./3).str (truncate = True)

1.3333333333333333
1.33333333333333

1.3333333333333333
1.33333333333333

len ((4./3).str (truncate = True)) - 1

sigdigs_bin2dec (24)

RR (e); N (RR (e), digits = 7)

2.71828182845905
2.718282

2.71828182845905
2.718282

ee = RealField(24) (e)
ee

2.71828

2.71828

N (ee, digits = 7)

Traceback (click to the left of this block for traceback)
...
TypeError: cannot approximate to a precision of 27 bits, use at most 24 bits

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_6.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TiAoZWUsIGRpZ2l0cyA9IDcp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpqkbP9v/___code___.py", line 3, in <module>
    exec compile(u'N (ee, digits = _sage_const_7 )
  File "", line 1, in <module>
    
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/misc/functional.py", line 1419, in numerical_approx
    return n(prec, algorithm=algorithm)
  File "sage/structure/element.pyx", line 850, in sage.structure.element.Element.numerical_approx (build/cythonized/sage/structure/element.c:7829)
  File "sage/arith/numerical_approx.pyx", line 65, in sage.arith.numerical_approx.numerical_approx_generic (build/cythonized/sage/arith/numerical_approx.c:2666)
TypeError: cannot approximate to a precision of 27 bits, use at most 24 bits

RealField(27) (e)

2.718282

2.718282

a = N (1, digits = 13)
parent (a)

Real Field with 47 bits of precision

Real Field with 47 bits of precision

strojna točnost (machine epsilon) — najmanji pozitivni broj $\epsilon_{\mathrm{m}}$ koji pribrojen broju 1 daje broj različit od 1: $1 + \epsilon_{\mathrm{m}} \ne 1$

najmanji broj veći od 1: $\bar{x} = (1,\!000\ldots001)_2 = 1 + 2^{-(p-1)} = 1 + \epsilon_{\mathrm{m}} \quad\Rightarrow\quad \epsilon_{\mathrm{m}} = 2^{-(p-1)}$

parent (1.)

Real Field with 53 bits of precision

Real Field with 53 bits of precision

2^(-(53-1))

1/4503599627370496

1/4503599627370496

1. + 2^(-(53-1)) == 1.

False

False

1. + 2^(-(54-1)) == 1.

True

True

N (2^(-(53-1)))

2.22044604925031e-16

2.22044604925031e-16

N (2^(-(54-1)))

1.11022302462516e-16

1.11022302462516e-16

def machine_epsilon (RF, prn = False) :
    one = eps = RF (1)
    nn = 0
    while one + eps != one :
        if prn : 
            print nn, ': ', eps
        eps /= 2
        nn += 1
    return 2 * eps

machine_epsilon (RR, prn = True)

0 :  1.00000000000000
1 :  0.500000000000000
2 :  0.250000000000000
3 :  0.125000000000000
4 :  0.0625000000000000
5 :  0.0312500000000000
6 :  0.0156250000000000
7 :  0.00781250000000000
8 :  0.00390625000000000
9 :  0.00195312500000000
10 :  0.000976562500000000
11 :  0.000488281250000000
12 :  0.000244140625000000
13 :  0.000122070312500000
14 :  0.0000610351562500000
15 :  0.0000305175781250000
16 :  0.0000152587890625000
17 :  7.62939453125000e-6
18 :  3.81469726562500e-6
19 :  1.90734863281250e-6
20 :  9.53674316406250e-7
21 :  4.76837158203125e-7
22 :  2.38418579101562e-7
23 :  1.19209289550781e-7
24 :  5.96046447753906e-8
25 :  2.98023223876953e-8
26 :  1.49011611938477e-8
27 :  7.45058059692383e-9
28 :  3.72529029846191e-9
29 :  1.86264514923096e-9
30 :  9.31322574615479e-10
31 :  4.65661287307739e-10
32 :  2.32830643653870e-10
33 :  1.16415321826935e-10
34 :  5.82076609134674e-11
35 :  2.91038304567337e-11
36 :  1.45519152283669e-11
37 :  7.27595761418343e-12
38 :  3.63797880709171e-12
39 :  1.81898940354586e-12
40 :  9.09494701772928e-13
41 :  4.54747350886464e-13
42 :  2.27373675443232e-13
43 :  1.13686837721616e-13
44 :  5.68434188608080e-14
45 :  2.84217094304040e-14
46 :  1.42108547152020e-14
47 :  7.10542735760100e-15
48 :  3.55271367880050e-15
49 :  1.77635683940025e-15
50 :  8.88178419700125e-16
51 :  4.44089209850063e-16
52 :  2.22044604925031e-16
2.22044604925031e-16

0 :  1.00000000000000
1 :  0.500000000000000
2 :  0.250000000000000
3 :  0.125000000000000
4 :  0.0625000000000000
5 :  0.0312500000000000
6 :  0.0156250000000000
7 :  0.00781250000000000
8 :  0.00390625000000000
9 :  0.00195312500000000
10 :  0.000976562500000000
11 :  0.000488281250000000
12 :  0.000244140625000000
13 :  0.000122070312500000
14 :  0.0000610351562500000
15 :  0.0000305175781250000
16 :  0.0000152587890625000
17 :  7.62939453125000e-6
18 :  3.81469726562500e-6
19 :  1.90734863281250e-6
20 :  9.53674316406250e-7
21 :  4.76837158203125e-7
22 :  2.38418579101562e-7
23 :  1.19209289550781e-7
24 :  5.96046447753906e-8
25 :  2.98023223876953e-8
26 :  1.49011611938477e-8
27 :  7.45058059692383e-9
28 :  3.72529029846191e-9
29 :  1.86264514923096e-9
30 :  9.31322574615479e-10
31 :  4.65661287307739e-10
32 :  2.32830643653870e-10
33 :  1.16415321826935e-10
34 :  5.82076609134674e-11
35 :  2.91038304567337e-11
36 :  1.45519152283669e-11
37 :  7.27595761418343e-12
38 :  3.63797880709171e-12
39 :  1.81898940354586e-12
40 :  9.09494701772928e-13
41 :  4.54747350886464e-13
42 :  2.27373675443232e-13
43 :  1.13686837721616e-13
44 :  5.68434188608080e-14
45 :  2.84217094304040e-14
46 :  1.42108547152020e-14
47 :  7.10542735760100e-15
48 :  3.55271367880050e-15
49 :  1.77635683940025e-15
50 :  8.88178419700125e-16
51 :  4.44089209850063e-16
52 :  2.22044604925031e-16
2.22044604925031e-16

machine_epsilon (RealField (14), True)

0 :  1.00
1 :  0.500
2 :  0.250
3 :  0.125
4 :  0.0625
5 :  0.0312
6 :  0.0156
7 :  0.00781
8 :  0.00391
9 :  0.00195
10 :  0.000977
11 :  0.000488
12 :  0.000244
13 :  0.000122
0.000122

0 :  1.00
1 :  0.500
2 :  0.250
3 :  0.125
4 :  0.0625
5 :  0.0312
6 :  0.0156
7 :  0.00781
8 :  0.00391
9 :  0.00195
10 :  0.000977
11 :  0.000488
12 :  0.000244
13 :  0.000122
0.000122

machine_epsilon (RealField(80))

1.6543612251060553497428e-24

1.6543612251060553497428e-24

2^(-(80-1))

1/604462909807314587353088

1/604462909807314587353088

RealField(80) (_)

1.6543612251060553497428e-24

1.6543612251060553497428e-24

jedinica na posljednjem mjestu (unit in the last place, ULP) — za (prikazivi) broj $x > 0$, razmak između $x$ i prvoga prikazivog većeg broja

(1.).ulp()

2.22044604925031e-16

2.22044604925031e-16

(10.).ulp()

1.77635683940025e-15

1.77635683940025e-15

(100.).ulp()

1.42108547152020e-14

1.42108547152020e-14

1. + (1.).ulp() == 1.

False

False

1. + (1.).ulp()/2 == 1.

True

True

100. + (100.).ulp() == 100.

False

False

100. + (100.).ulp()/2 == 100.

True

True

(1.).ulp() == machine_epsilon (parent (1.))

True

True

2. + (1.).ulp() == 2.

True

True

2. + (2.).ulp() == 2.

False

False

32. + (16.).ulp() == 32.

True

True

32. + (32.).ulp() == 32.

False

False

ULP za brojeve $2^{i_b},\, 2^{i_b+1}, ... ,\, 2^{i_e-1}$:

def print_ulps (RF, ib, ie) :
    two = RF (2)
    for i in xrange (ib, ie) :
        x = two^i
        print i, ' : ', x, ' : ', x.ulp()

print_ulps (RR, -10, 11)

-10  :  0.000976562500000000  :  2.16840434497101e-19
-9  :  0.00195312500000000  :  4.33680868994202e-19
-8  :  0.00390625000000000  :  8.67361737988404e-19
-7  :  0.00781250000000000  :  1.73472347597681e-18
-6  :  0.0156250000000000  :  3.46944695195361e-18
-5  :  0.0312500000000000  :  6.93889390390723e-18
-4  :  0.0625000000000000  :  1.38777878078145e-17
-3  :  0.125000000000000  :  2.77555756156289e-17
-2  :  0.250000000000000  :  5.55111512312578e-17
-1  :  0.500000000000000  :  1.11022302462516e-16
0  :  1.00000000000000  :  2.22044604925031e-16
1  :  2.00000000000000  :  4.44089209850063e-16
2  :  4.00000000000000  :  8.88178419700125e-16
3  :  8.00000000000000  :  1.77635683940025e-15
4  :  16.0000000000000  :  3.55271367880050e-15
5  :  32.0000000000000  :  7.10542735760100e-15
6  :  64.0000000000000  :  1.42108547152020e-14
7  :  128.000000000000  :  2.84217094304040e-14
8  :  256.000000000000  :  5.68434188608080e-14
9  :  512.000000000000  :  1.13686837721616e-13
10  :  1024.00000000000  :  2.27373675443232e-13

-10  :  0.000976562500000000  :  2.16840434497101e-19
-9  :  0.00195312500000000  :  4.33680868994202e-19
-8  :  0.00390625000000000  :  8.67361737988404e-19
-7  :  0.00781250000000000  :  1.73472347597681e-18
-6  :  0.0156250000000000  :  3.46944695195361e-18
-5  :  0.0312500000000000  :  6.93889390390723e-18
-4  :  0.0625000000000000  :  1.38777878078145e-17
-3  :  0.125000000000000  :  2.77555756156289e-17
-2  :  0.250000000000000  :  5.55111512312578e-17
-1  :  0.500000000000000  :  1.11022302462516e-16
0  :  1.00000000000000  :  2.22044604925031e-16
1  :  2.00000000000000  :  4.44089209850063e-16
2  :  4.00000000000000  :  8.88178419700125e-16
3  :  8.00000000000000  :  1.77635683940025e-15
4  :  16.0000000000000  :  3.55271367880050e-15
5  :  32.0000000000000  :  7.10542735760100e-15
6  :  64.0000000000000  :  1.42108547152020e-14
7  :  128.000000000000  :  2.84217094304040e-14
8  :  256.000000000000  :  5.68434188608080e-14
9  :  512.000000000000  :  1.13686837721616e-13
10  :  1024.00000000000  :  2.27373675443232e-13

- ako je $x = 2^k$, onda je $\mathrm{ulp}\,(x) = \epsilon_{\mathrm{m}}\!\cdot 2^k$

print_ulps (RealField(80), -10, 11)

-10  :  0.00097656250000000000000000  :  1.6155871338926321774832e-27
-9  :  0.0019531250000000000000000  :  3.2311742677852643549664e-27
-8  :  0.0039062500000000000000000  :  6.4623485355705287099329e-27
-7  :  0.0078125000000000000000000  :  1.2924697071141057419866e-26
-6  :  0.015625000000000000000000  :  2.5849394142282114839732e-26
-5  :  0.031250000000000000000000  :  5.1698788284564229679463e-26
-4  :  0.062500000000000000000000  :  1.0339757656912845935893e-25
-3  :  0.12500000000000000000000  :  2.0679515313825691871785e-25
-2  :  0.25000000000000000000000  :  4.1359030627651383743570e-25
-1  :  0.50000000000000000000000  :  8.2718061255302767487141e-25
0  :  1.0000000000000000000000  :  1.6543612251060553497428e-24
1  :  2.0000000000000000000000  :  3.3087224502121106994856e-24
2  :  4.0000000000000000000000  :  6.6174449004242213989713e-24
3  :  8.0000000000000000000000  :  1.3234889800848442797943e-23
4  :  16.000000000000000000000  :  2.6469779601696885595885e-23
5  :  32.000000000000000000000  :  5.2939559203393771191770e-23
6  :  64.000000000000000000000  :  1.0587911840678754238354e-22
7  :  128.00000000000000000000  :  2.1175823681357508476708e-22
8  :  256.00000000000000000000  :  4.2351647362715016953416e-22
9  :  512.00000000000000000000  :  8.4703294725430033906832e-22
10  :  1024.0000000000000000000  :  1.6940658945086006781366e-21

-10  :  0.00097656250000000000000000  :  1.6155871338926321774832e-27
-9  :  0.0019531250000000000000000  :  3.2311742677852643549664e-27
-8  :  0.0039062500000000000000000  :  6.4623485355705287099329e-27
-7  :  0.0078125000000000000000000  :  1.2924697071141057419866e-26
-6  :  0.015625000000000000000000  :  2.5849394142282114839732e-26
-5  :  0.031250000000000000000000  :  5.1698788284564229679463e-26
-4  :  0.062500000000000000000000  :  1.0339757656912845935893e-25
-3  :  0.12500000000000000000000  :  2.0679515313825691871785e-25
-2  :  0.25000000000000000000000  :  4.1359030627651383743570e-25
-1  :  0.50000000000000000000000  :  8.2718061255302767487141e-25
0  :  1.0000000000000000000000  :  1.6543612251060553497428e-24
1  :  2.0000000000000000000000  :  3.3087224502121106994856e-24
2  :  4.0000000000000000000000  :  6.6174449004242213989713e-24
3  :  8.0000000000000000000000  :  1.3234889800848442797943e-23
4  :  16.000000000000000000000  :  2.6469779601696885595885e-23
5  :  32.000000000000000000000  :  5.2939559203393771191770e-23
6  :  64.000000000000000000000  :  1.0587911840678754238354e-22
7  :  128.00000000000000000000  :  2.1175823681357508476708e-22
8  :  256.00000000000000000000  :  4.2351647362715016953416e-22
9  :  512.00000000000000000000  :  8.4703294725430033906832e-22
10  :  1024.0000000000000000000  :  1.6940658945086006781366e-21

print_ulps (RealField(80), 70, 81)

70  :  1.1805916207174113034240e21  :  0.0019531250000000000000000
71  :  2.3611832414348226068480e21  :  0.0039062500000000000000000
72  :  4.7223664828696452136960e21  :  0.0078125000000000000000000
73  :  9.4447329657392904273920e21  :  0.015625000000000000000000
74  :  1.8889465931478580854784e22  :  0.031250000000000000000000
75  :  3.7778931862957161709568e22  :  0.062500000000000000000000
76  :  7.5557863725914323419136e22  :  0.12500000000000000000000
77  :  1.5111572745182864683827e23  :  0.25000000000000000000000
78  :  3.0223145490365729367654e23  :  0.50000000000000000000000
79  :  6.0446290980731458735309e23  :  1.0000000000000000000000
80  :  1.2089258196146291747062e24  :  2.0000000000000000000000

70  :  1.1805916207174113034240e21  :  0.0019531250000000000000000
71  :  2.3611832414348226068480e21  :  0.0039062500000000000000000
72  :  4.7223664828696452136960e21  :  0.0078125000000000000000000
73  :  9.4447329657392904273920e21  :  0.015625000000000000000000
74  :  1.8889465931478580854784e22  :  0.031250000000000000000000
75  :  3.7778931862957161709568e22  :  0.062500000000000000000000
76  :  7.5557863725914323419136e22  :  0.12500000000000000000000
77  :  1.5111572745182864683827e23  :  0.25000000000000000000000
78  :  3.0223145490365729367654e23  :  0.50000000000000000000000
79  :  6.0446290980731458735309e23  :  1.0000000000000000000000
80  :  1.2089258196146291747062e24  :  2.0000000000000000000000

print_ulps (RR, 50, 71)

50  :  1.12589990684262e15  :  0.250000000000000
51  :  2.25179981368525e15  :  0.500000000000000
52  :  4.50359962737050e15  :  1.00000000000000
53  :  9.00719925474099e15  :  2.00000000000000
54  :  1.80143985094820e16  :  4.00000000000000
55  :  3.60287970189640e16  :  8.00000000000000
56  :  7.20575940379279e16  :  16.0000000000000
57  :  1.44115188075856e17  :  32.0000000000000
58  :  2.88230376151712e17  :  64.0000000000000
59  :  5.76460752303423e17  :  128.000000000000
60  :  1.15292150460685e18  :  256.000000000000
61  :  2.30584300921369e18  :  512.000000000000
62  :  4.61168601842739e18  :  1024.00000000000
63  :  9.22337203685478e18  :  2048.00000000000
64  :  1.84467440737096e19  :  4096.00000000000
65  :  3.68934881474191e19  :  8192.00000000000
66  :  7.37869762948382e19  :  16384.0000000000
67  :  1.47573952589676e20  :  32768.0000000000
68  :  2.95147905179353e20  :  65536.0000000000
69  :  5.90295810358706e20  :  131072.000000000
70  :  1.18059162071741e21  :  262144.000000000

50  :  1.12589990684262e15  :  0.250000000000000
51  :  2.25179981368525e15  :  0.500000000000000
52  :  4.50359962737050e15  :  1.00000000000000
53  :  9.00719925474099e15  :  2.00000000000000
54  :  1.80143985094820e16  :  4.00000000000000
55  :  3.60287970189640e16  :  8.00000000000000
56  :  7.20575940379279e16  :  16.0000000000000
57  :  1.44115188075856e17  :  32.0000000000000
58  :  2.88230376151712e17  :  64.0000000000000
59  :  5.76460752303423e17  :  128.000000000000
60  :  1.15292150460685e18  :  256.000000000000
61  :  2.30584300921369e18  :  512.000000000000
62  :  4.61168601842739e18  :  1024.00000000000
63  :  9.22337203685478e18  :  2048.00000000000
64  :  1.84467440737096e19  :  4096.00000000000
65  :  3.68934881474191e19  :  8192.00000000000
66  :  7.37869762948382e19  :  16384.0000000000
67  :  1.47573952589676e20  :  32768.0000000000
68  :  2.95147905179353e20  :  65536.0000000000
69  :  5.90295810358706e20  :  131072.000000000
70  :  1.18059162071741e21  :  262144.000000000

print_ulps (RealField (14), 0, 11)

0  :  1.00  :  0.000122
1  :  2.00  :  0.000244
2  :  4.00  :  0.000488
3  :  8.00  :  0.000977
4  :  16.0  :  0.00195
5  :  32.0  :  0.00391
6  :  64.0  :  0.00781
7  :  128.  :  0.0156
8  :  256.  :  0.0312
9  :  512.  :  0.0625
10  :  1020.  :  0.125

0  :  1.00  :  0.000122
1  :  2.00  :  0.000244
2  :  4.00  :  0.000488
3  :  8.00  :  0.000977
4  :  16.0  :  0.00195
5  :  32.0  :  0.00391
6  :  64.0  :  0.00781
7  :  128.  :  0.0156
8  :  256.  :  0.0312
9  :  512.  :  0.0625
10  :  1020.  :  0.125

ULP za brojeve $x,\, x+dx,\, x+2\,dx, ... ,\, x+(n-1)\,dx$:

def print_ulps2 (x, dx, nn) :  
    for i in xrange (nn) :
        print i, ' : ', x,' : ', x.ulp()
        x += dx

print_ulps2 (1., 0.1, 11)

0  :  1.00000000000000  :  2.22044604925031e-16
1  :  1.10000000000000  :  2.22044604925031e-16
2  :  1.20000000000000  :  2.22044604925031e-16
3  :  1.30000000000000  :  2.22044604925031e-16
4  :  1.40000000000000  :  2.22044604925031e-16
5  :  1.50000000000000  :  2.22044604925031e-16
6  :  1.60000000000000  :  2.22044604925031e-16
7  :  1.70000000000000  :  2.22044604925031e-16
8  :  1.80000000000000  :  2.22044604925031e-16
9  :  1.90000000000000  :  2.22044604925031e-16
10  :  2.00000000000000  :  4.44089209850063e-16

0  :  1.00000000000000  :  2.22044604925031e-16
1  :  1.10000000000000  :  2.22044604925031e-16
2  :  1.20000000000000  :  2.22044604925031e-16
3  :  1.30000000000000  :  2.22044604925031e-16
4  :  1.40000000000000  :  2.22044604925031e-16
5  :  1.50000000000000  :  2.22044604925031e-16
6  :  1.60000000000000  :  2.22044604925031e-16
7  :  1.70000000000000  :  2.22044604925031e-16
8  :  1.80000000000000  :  2.22044604925031e-16
9  :  1.90000000000000  :  2.22044604925031e-16
10  :  2.00000000000000  :  4.44089209850063e-16

print_ulps2 (1., 1., 15)

0  :  1.00000000000000  :  2.22044604925031e-16
1  :  2.00000000000000  :  4.44089209850063e-16
2  :  3.00000000000000  :  4.44089209850063e-16
3  :  4.00000000000000  :  8.88178419700125e-16
4  :  5.00000000000000  :  8.88178419700125e-16
5  :  6.00000000000000  :  8.88178419700125e-16
6  :  7.00000000000000  :  8.88178419700125e-16
7  :  8.00000000000000  :  1.77635683940025e-15
8  :  9.00000000000000  :  1.77635683940025e-15
9  :  10.0000000000000  :  1.77635683940025e-15
10  :  11.0000000000000  :  1.77635683940025e-15
11  :  12.0000000000000  :  1.77635683940025e-15
12  :  13.0000000000000  :  1.77635683940025e-15
13  :  14.0000000000000  :  1.77635683940025e-15
14  :  15.0000000000000  :  1.77635683940025e-15

0  :  1.00000000000000  :  2.22044604925031e-16
1  :  2.00000000000000  :  4.44089209850063e-16
2  :  3.00000000000000  :  4.44089209850063e-16
3  :  4.00000000000000  :  8.88178419700125e-16
4  :  5.00000000000000  :  8.88178419700125e-16
5  :  6.00000000000000  :  8.88178419700125e-16
6  :  7.00000000000000  :  8.88178419700125e-16
7  :  8.00000000000000  :  1.77635683940025e-15
8  :  9.00000000000000  :  1.77635683940025e-15
9  :  10.0000000000000  :  1.77635683940025e-15
10  :  11.0000000000000  :  1.77635683940025e-15
11  :  12.0000000000000  :  1.77635683940025e-15
12  :  13.0000000000000  :  1.77635683940025e-15
13  :  14.0000000000000  :  1.77635683940025e-15
14  :  15.0000000000000  :  1.77635683940025e-15

- ako je $x = m\cdot 2^k$, gdje je $m$ normalizirana mantisa, onda je $\mathrm{ulp}\,(x) = \epsilon_{\mathrm{m}}\!\cdot 2^k$

print_ulps2 (RealField(80) (2.), RealField(80) (0.1), 22)

0  :  2.0000000000000000000000  :  3.3087224502121106994856e-24
1  :  2.1000000000000000000000  :  3.3087224502121106994856e-24
2  :  2.2000000000000000000000  :  3.3087224502121106994856e-24
3  :  2.3000000000000000000000  :  3.3087224502121106994856e-24
4  :  2.4000000000000000000000  :  3.3087224502121106994856e-24
5  :  2.5000000000000000000000  :  3.3087224502121106994856e-24
6  :  2.6000000000000000000000  :  3.3087224502121106994856e-24
7  :  2.7000000000000000000000  :  3.3087224502121106994856e-24
8  :  2.8000000000000000000000  :  3.3087224502121106994856e-24
9  :  2.9000000000000000000000  :  3.3087224502121106994856e-24
10  :  3.0000000000000000000000  :  3.3087224502121106994856e-24
11  :  3.1000000000000000000000  :  3.3087224502121106994856e-24
12  :  3.2000000000000000000000  :  3.3087224502121106994856e-24
13  :  3.3000000000000000000000  :  3.3087224502121106994856e-24
14  :  3.4000000000000000000000  :  3.3087224502121106994856e-24
15  :  3.5000000000000000000000  :  3.3087224502121106994856e-24
16  :  3.6000000000000000000000  :  3.3087224502121106994856e-24
17  :  3.7000000000000000000000  :  3.3087224502121106994856e-24
18  :  3.8000000000000000000000  :  3.3087224502121106994856e-24
19  :  3.9000000000000000000000  :  3.3087224502121106994856e-24
20  :  4.0000000000000000000000  :  3.3087224502121106994856e-24
21  :  4.1000000000000000000000  :  6.6174449004242213989713e-24

0  :  2.0000000000000000000000  :  3.3087224502121106994856e-24
1  :  2.1000000000000000000000  :  3.3087224502121106994856e-24
2  :  2.2000000000000000000000  :  3.3087224502121106994856e-24
3  :  2.3000000000000000000000  :  3.3087224502121106994856e-24
4  :  2.4000000000000000000000  :  3.3087224502121106994856e-24
5  :  2.5000000000000000000000  :  3.3087224502121106994856e-24
6  :  2.6000000000000000000000  :  3.3087224502121106994856e-24
7  :  2.7000000000000000000000  :  3.3087224502121106994856e-24
8  :  2.8000000000000000000000  :  3.3087224502121106994856e-24
9  :  2.9000000000000000000000  :  3.3087224502121106994856e-24
10  :  3.0000000000000000000000  :  3.3087224502121106994856e-24
11  :  3.1000000000000000000000  :  3.3087224502121106994856e-24
12  :  3.2000000000000000000000  :  3.3087224502121106994856e-24
13  :  3.3000000000000000000000  :  3.3087224502121106994856e-24
14  :  3.4000000000000000000000  :  3.3087224502121106994856e-24
15  :  3.5000000000000000000000  :  3.3087224502121106994856e-24
16  :  3.6000000000000000000000  :  3.3087224502121106994856e-24
17  :  3.7000000000000000000000  :  3.3087224502121106994856e-24
18  :  3.8000000000000000000000  :  3.3087224502121106994856e-24
19  :  3.9000000000000000000000  :  3.3087224502121106994856e-24
20  :  4.0000000000000000000000  :  3.3087224502121106994856e-24
21  :  4.1000000000000000000000  :  6.6174449004242213989713e-24

zbrajanje „realnih” brojeva nije asocijativno:

u = (1.).ulp() / 2

(1. + u) + u == 1. + (u + u)

False

False

1. + (u + u) == 1.

False

False

print 1. + (u + u)
print (1. + (u + u)).str()

1.00000000000000
1.0000000000000002

1.00000000000000
1.0000000000000002

(1. + u) + u == 1.

True

True

print (1. + u) + u
print ((1. + u) + u).str()

1.00000000000000
1.0000000000000000

1.00000000000000
1.0000000000000000

j = 1.
for i in xrange (1000000) :
    j += u;
j == 1.; j

True
1.00000000000000

True
1.00000000000000

j = 1.
k = 0.
for i in xrange (2) :
    k += u;
j += k
j == 1.; j

False
1.00000000000000

False
1.00000000000000

zbrajanje brojeva tipa RealField(14):

$3 \,+\, 2\;=\; (11000000000000)_2 \times 2^{-12} \,+\, (10000000000000)_2 \times 2^{-12}\;:$

$\phantom{+}\;\:(\phantom{1}11000000000000\;)_2 \times 2^{-12}$

$+\;\:(\phantom{1}10000000000000\;)_2 \times 2^{-12}$

$=\kern0.1625ex\:(101000000000000\:)_2 \times 2^{-12}$

$=\kern0.1625ex\:(\phantom{1}10100000000000\;)_2 \times 2^{-11}$ $\ldots$ normalizacija

$3 \,+\, \dfrac{3}{4} \;=\; (11000000000000)_2 \times 2^{-12} \,+\, (11000000000000)_2 \times 2^{-14}\;:$

$\phantom{+}\;\:(\;11000000000000\;)_2 \times 2^{-12}$

$+\;\:(\;00110000000000\;)_2 \times 2^{-12}$ $\ldots$ pomicanje mantise $-12 - (-14) = 2$ mjesta udesno

$=\kern0.1625ex\:(\;11110000000000\;)_2 \times 2^{-12}$

$1 \, +\, \dfrac{1}{2}\,\mathrm{ulp} (1) \;=\; (10000000000000)_2 \times 2^{-13} \,+\, (10000000000000)_2 \times 2^{-27}\;:$

$\phantom{+}\;\:(\;10000000000000\phantom{\Big|1}\;)_2 \times 2^{-13}$

$+\;\:(\;00000000000000\Big|1\;)_2 \times 2^{-13}$ $\ldots$ pomicanje mantise $-13 - (-27) = 14$ mjesta udesno

$=\kern0.1625ex\:(\;10000000000000\Big|1\;)_2 \times 2^{-13}$

$\approx\kern0.1625ex\:(\;10000000000000\phantom{\Big|1}\;)_2 \times 2^{-13}$

$ \dfrac{1}{2}\,\mathrm{ulp} (1)\, +\, \dfrac{1}{2}\,\mathrm{ulp} (1) \;=\; (10000000000000)_2 \times 2^{-27} \,+\, (10000000000000)_2 \times 2^{-27}\;:$

$\phantom{+}\;\:(\phantom{1}10000000000000\;)_2 \times 2^{-27}$

$+\;\:(\phantom{1}10000000000000\;)_2 \times 2^{-27}$

$=\kern0.1625ex\:(100000000000000\:)_2 \times 2^{-27}$

$=\kern0.1625ex\:(\phantom{1}10000000000000\;)_2 \times 2^{-26}$

$1 \, +\, \mathrm{ulp} (1) \;=\; (10000000000000)_2 \times 2^{-13} \,+\, (10000000000000)_2 \times 2^{-26}\;:$

$\phantom{+}\;\:(\;11000000000000\;)_2 \times 2^{-13}$

$+\;\:(\;00000000000001\;)_2 \times 2^{-13}$ $\ldots$ pomicanje mantise $-13 - (-26) = 13$ mjesta udesno

$=\kern0.1625ex\:(\;10000000000001\;)_2 \times 2^{-13}$

zaokruživanja (rounding, RND)

prema najbližem, to nearest, N (podrazumijeva se):

a = RealField(14, rnd = 'RNDN') (1/3)
b = RealField(14, rnd = 'RNDN') (2/3) 
a, b, -a, -b

(0.333, 0.667, -0.333, -0.667)

(0.333, 0.667, -0.333, -0.667)

naniže (prema $-\infty$), down, D:

a = RealField(14, rnd = 'RNDD') (1/3)
b = RealField(14, rnd = 'RNDD') (2/3) 
a, b, -a, -b

(0.333, 0.666, -0.334, -0.667)

(0.333, 0.666, -0.334, -0.667)

naviše (prema $+\infty$), up, U:

a = RealField(14, rnd = 'RNDU') (1/3)
b = RealField(14, rnd = 'RNDU') (2/3) 
a, b, -a, -b

(0.334, 0.667, -0.333, -0.666)

(0.334, 0.667, -0.333, -0.666)

prema nuli, towards zero, Z:

a = RealField(14, rnd = 'RNDZ') (1/3)
b = RealField(14, rnd = 'RNDZ') (2/3) 
a, b, -a, -b

(0.333, 0.666, -0.333, -0.666)

(0.333, 0.666, -0.333, -0.666)

od nule, away from zero, A:

a = RealField(14, rnd = 'RNDA') (1/3)
b = RealField(14, rnd = 'RNDA') (2/3) 
a, b, -a, -b

(0.334, 0.667, -0.334, -0.667)

(0.334, 0.667, -0.334, -0.667)

zbrajanja i zaokruživanja:

(RealField (rnd = 'RNDN') (1. + u)) + u == 1., (RealField (rnd = 'RNDN') (-1. - u)) - u == -1.

(True, True)

(True, True)

(RealField (rnd = 'RNDD') (1. + u)) + u == 1., (RealField (rnd = 'RNDD') (-1. - u)) - u == -1.

(True, False)

(True, False)

(RealField (rnd = 'RNDU') (1. + u)) + u == 1., (RealField (rnd = 'RNDU') (-1. - u)) - u == -1.

(False, True)

(False, True)

(RealField (rnd = 'RNDZ') (1. + u)) + u == 1., (RealField (rnd = 'RNDZ') (-1. - u)) - u == -1.

(True, True)

(True, True)

(RealField (rnd = 'RNDA') (1. + u)) + u == 1., (RealField (rnd = 'RNDA') (-1. - u)) - u == -1.

(False, False)

(False, False)

(apsolutna) greška zaokruživanja:

$|\bar{x} \,-\, x| \:<\: \mathrm{ulp}\,(\bar{x})$ za D, U, Z, A

$|\bar{x} \,-\, x| \:<\: \dfrac{1}{2}\,\mathrm{ulp}\,(\bar{x})$ za N

„intervalna” aritmetika:

pi_d = RealField(rnd = 'RNDD') (pi)
e_d = RealField(rnd = 'RNDD') (e)
s_d = sqrt (pi_d - e_d)
pi_d, e_d, s_d

(3.14159265358979, 2.71828182845904, 0.650623412682596)

(3.14159265358979, 2.71828182845904, 0.650623412682596)

pi_u = RealField(rnd = 'RNDU') (pi)
e_u = RealField(rnd = 'RNDU') (e)
s_u = sqrt (pi_u - e_u)
pi_u, e_u, s_u

(3.14159265358980, 2.71828182845905, 0.650623412682597)

(3.14159265358980, 2.71828182845905, 0.650623412682597)

... realni broj $\,\sqrt{\pi - r}\,$ leži u intervalu između „realnih” brojeva $s_d$ i $s_u$

ispitivanje jednakosti „realnih” brojeva

s = 0.
for i in xrange (10) : 
    s += 0.1
s

1.00000000000000

1.00000000000000

s == 1.

False

False

s = 0.
while (s != 1.) :
    s += 0.1
s

^C
Traceback (click to the left of this block for traceback)
...
__SAGE__

^C
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_3.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cyA9IDAuCndoaWxlIChzICE9IDEuKSA6CiAgICBzICs9IDAuMQpz"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpQnoHQZ/___code___.py", line 5, in <module>
    s += _sage_const_0p1 
  File "src/cysignals/signals.pyx", line 320, in cysignals.signals.python_check_interrupt
KeyboardInterrupt
__SAGE__

s

2.64211950131177e6

2.64211950131177e6

s = 0.
for i in xrange (12) :
    s += 0.1
    s.str (truncate = False)

'0.10000000000000001'
'0.20000000000000001'
'0.30000000000000004'
'0.40000000000000002'
'0.50000000000000000'
'0.59999999999999998'
'0.69999999999999996'
'0.79999999999999993'
'0.89999999999999991'
'0.99999999999999989'
'1.0999999999999999'
'1.2000000000000000'

'0.10000000000000001'
'0.20000000000000001'
'0.30000000000000004'
'0.40000000000000002'
'0.50000000000000000'
'0.59999999999999998'
'0.69999999999999996'
'0.79999999999999993'
'0.89999999999999991'
'0.99999999999999989'
'1.0999999999999999'
'1.2000000000000000'

$\dfrac{1}{10} = 0,\!1 = (0,\!00011001100110011\ldots)_2 = (0,\!0\overline{0011})_2$

a53 = RR (1/10)
a1325 = RealField(1325) (1/10)
print a53, '\n\n', a1325

0.100000000000000 

0.100000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000

0.100000000000000 

0.10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

print a53.str (base = 2), '\n'
print a1325.str (base = 2)

0.00011001100110011001100110011001100110011001100110011010 

0.000110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011001100110011001100110011001100110\
01100110011001100110011001100110011001100110011010

0.00011001100110011001100110011001100110011001100110011010 

0.00011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011010

print a53.str(), '\n'
print a1325.str()

0.10000000000000001 

0.100000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
01

0.10000000000000001 

0.1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

RealField(1325) (a53)

0.100000000000000005551115123125782702118158340454101562500000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000000000000000

0.10000000000000000555111512312578270211815834045410156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

N (a53, 1325)

Traceback (click to the left of this block for traceback)
...
TypeError: cannot approximate to a precision of 1325 bits, use at most 53 bits

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_87.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TiAoYTUzLCAxMzI1KQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpqZQ98Y/___code___.py", line 3, in <module>
    exec compile(u'N (a53, _sage_const_1325 )
  File "", line 1, in <module>
    
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/misc/functional.py", line 1419, in numerical_approx
    return n(prec, algorithm=algorithm)
  File "sage/structure/element.pyx", line 850, in sage.structure.element.Element.numerical_approx (build/cythonized/sage/structure/element.c:7829)
  File "sage/arith/numerical_approx.pyx", line 65, in sage.arith.numerical_approx.numerical_approx_generic (build/cythonized/sage/arith/numerical_approx.c:2666)
TypeError: cannot approximate to a precision of 1325 bits, use at most 53 bits

A = matrix ([[4., -1., -1., 0., 0.], 
             [-1., 4., -1., -1., 0.], 
             [-1., -1., 4., -1., -1.], 
             [0., -1., -1., 4., -1.], 
             [0., 0., -1., -1., 4.]])
show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr} 4.00000000000000 & -1.00000000000000 & -1.00000000000000 & 0.000000000000000 & 0.000000000000000 \\ -1.00000000000000 & 4.00000000000000 & -1.00000000000000 & -1.00000000000000 & 0.000000000000000 \\ -1.00000000000000 & -1.00000000000000 & 4.00000000000000 & -1.00000000000000 & -1.00000000000000 \\ 0.000000000000000 & -1.00000000000000 & -1.00000000000000 & 4.00000000000000 & -1.00000000000000 \\ 0.000000000000000 & 0.000000000000000 & -1.00000000000000 & -1.00000000000000 & 4.00000000000000 \end{array}\right)$

Ai = A.inverse()
show (Ai)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr} 0.323886639676113 & 0.141700404858300 & 0.153846153846154 & 0.0890688259109312 & 0.0607287449392712 \\ 0.141700404858300 & 0.374493927125506 & 0.192307692307692 & 0.163967611336032 & 0.0890688259109312 \\ 0.153846153846154 & 0.192307692307692 & 0.423076923076923 & 0.192307692307692 & 0.153846153846154 \\ 0.0890688259109312 & 0.163967611336032 & 0.192307692307692 & 0.374493927125506 & 0.141700404858300 \\ 0.0607287449392712 & 0.0890688259109312 & 0.153846153846154 & 0.141700404858300 & 0.323886639676113 \end{array}\right)$

za inverznu bi matricu trebalo vrijediti $\mathbf{A}\,\mathbf{A}^{\!-1} = \mathbf{A}^{\!-1}\mathbf{A} = \mathbf{I}$, ali...

A * Ai == identity_matrix (RR, 5)

False

False

Ai * A == identity_matrix (RR, 5)

False

False

A * Ai == Ai * A

False

False

... naime, kako „realna” aritmetika na računalu nije matematička realna aritmetika beskonačne točnosti, komponente matrice $\mathbf{A}\,\mathbf{A}^{\!-1}$ koje bi trebale biti nule, nisu nule:

show (A * Ai)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr} 1.00000000000000 & -2.77555756156289 \times 10^{-17} & 0.000000000000000 & -2.77555756156289 \times 10^{-17} & -2.77555756156289 \times 10^{-17} \\ -5.55111512312578 \times 10^{-17} & 1.00000000000000 & -1.11022302462516 \times 10^{-16} & -5.55111512312578 \times 10^{-17} & 0.000000000000000 \\ 6.93889390390723 \times 10^{-18} & -8.32667268468867 \times 10^{-17} & 1.00000000000000 & -2.77555756156289 \times 10^{-17} & 0.000000000000000 \\ -2.08166817117217 \times 10^{-17} & 8.32667268468867 \times 10^{-17} & 2.77555756156289 \times 10^{-17} & 1.00000000000000 & 0.000000000000000 \\ -2.77555756156289 \times 10^{-17} & 0.000000000000000 & 0.000000000000000 & -1.11022302462516 \times 10^{-16} & 1.00000000000000 \end{array}\right)$

... štoviše, nisu ni sve „jedinice” jedinice, iako su ispisane kao jedinice:

print (1.).exact_rational()
print 
print (A * Ai)[0,0].exact_rational()
print (A * Ai)[1,1].exact_rational()
print (A * Ai)[2,2].exact_rational()
print (A * Ai)[3,3].exact_rational()
print (A * Ai)[4,4].exact_rational()

1

4503599627370497/4503599627370496
1
9007199254740991/9007199254740992
9007199254740991/9007199254740992
4503599627370495/4503599627370496

1

4503599627370497/4503599627370496
1
9007199254740991/9007199254740992
9007199254740991/9007199254740992
4503599627370495/4503599627370496

print (A * Ai)[0,0].str()
print (A * Ai)[1,1].str()
print (A * Ai)[2,2].str()
print (A * Ai)[3,3].str()
print (A * Ai)[4,4].str()

1.0000000000000002
1.0000000000000000
0.99999999999999989
0.99999999999999989
0.99999999999999978

1.0000000000000002
1.0000000000000000
0.99999999999999989
0.99999999999999989
0.99999999999999978

neki (ali nikako svi) ispravni načini ispitivanja „jednakosti”:

tolerance = sqrt ((1.0).ulp())  # recimo...
tolerance

1.49011611938477e-8

1.49011611938477e-8

(A * Ai - identity_matrix (RR, 5)).apply_map (lambda x : abs (x) < tolerance)

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

(Ai * A - identity_matrix (RR, 5)).apply_map (lambda x : abs (x) < tolerance)

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

(A * Ai - Ai * A).apply_map (lambda x : abs (x) < tolerance)

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

... ili, varijacija na temu:

(A * Ai - identity_matrix (RR, 5)).find (lambda x : abs (x) < tolerance)

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]
[1 1 1 1 1]

(A * Ai - identity_matrix (RR, 5)).find (lambda x : abs (x) < tolerance, indices = True)

{(0, 0): 2.22044604925031e-16,
 (0, 1): -2.77555756156289e-17,
 (0, 2): 0.000000000000000,
 (0, 3): -2.77555756156289e-17,
 (0, 4): -2.77555756156289e-17,
 (1, 0): -5.55111512312578e-17,
 (1, 1): 0.000000000000000,
 (1, 2): -1.11022302462516e-16,
 (1, 3): -5.55111512312578e-17,
 (1, 4): 0.000000000000000,
 (2, 0): 6.93889390390723e-18,
 (2, 1): -8.32667268468867e-17,
 (2, 2): -1.11022302462516e-16,
 (2, 3): -2.77555756156289e-17,
 (2, 4): 0.000000000000000,
 (3, 0): -2.08166817117217e-17,
 (3, 1): 8.32667268468867e-17,
 (3, 2): 2.77555756156289e-17,
 (3, 3): -1.11022302462516e-16,
 (3, 4): 0.000000000000000,
 (4, 0): -2.77555756156289e-17,
 (4, 1): 0.000000000000000,
 (4, 2): 0.000000000000000,
 (4, 3): -1.11022302462516e-16,
 (4, 4): -2.22044604925031e-16}

{(0, 0): 2.22044604925031e-16,
 (0, 1): -2.77555756156289e-17,
 (0, 2): 0.000000000000000,
 (0, 3): -2.77555756156289e-17,
 (0, 4): -2.77555756156289e-17,
 (1, 0): -5.55111512312578e-17,
 (1, 1): 0.000000000000000,
 (1, 2): -1.11022302462516e-16,
 (1, 3): -5.55111512312578e-17,
 (1, 4): 0.000000000000000,
 (2, 0): 6.93889390390723e-18,
 (2, 1): -8.32667268468867e-17,
 (2, 2): -1.11022302462516e-16,
 (2, 3): -2.77555756156289e-17,
 (2, 4): 0.000000000000000,
 (3, 0): -2.08166817117217e-17,
 (3, 1): 8.32667268468867e-17,
 (3, 2): 2.77555756156289e-17,
 (3, 3): -1.11022302462516e-16,
 (3, 4): 0.000000000000000,
 (4, 0): -2.77555756156289e-17,
 (4, 1): 0.000000000000000,
 (4, 2): 0.000000000000000,
 (4, 3): -1.11022302462516e-16,
 (4, 4): -2.22044604925031e-16}

len ((A * Ai - identity_matrix (RR, 5)).find (lambda x : abs (x) < tolerance, indices = True))

len ((A * Ai - identity_matrix (RR, 5)).find (lambda x : abs (x) < tolerance, indices = True)) == A.nrows() * A.ncols()

True

True

(A * Ai - identity_matrix (RR, 5)).find (lambda x : abs (x) >= tolerance, indices = True)

{}

{}

len ((A * Ai - identity_matrix (RR, 5)).find (lambda x : abs (x) >= tolerance, indices = True))

len ((A * Ai - identity_matrix (RR, 5)).find (lambda x : abs (x) >= tolerance, indices = True)) == 0

True

True

... ili:

norm (A * Ai - identity_matrix (RR, 5)) < tolerance       #  the Euclidean norm

True

True

norm (Ai * A - identity_matrix (RR, 5)) < tolerance

True

True

... ili, varijacije:

(Ai * A - identity_matrix (RR, 5)).norm (1) < tolerance    # the largest column-sum norm

True

True

(Ai * A - identity_matrix (RR, 5)).norm (oo) < tolerance   # the largest row-sum norm

True

True

(A * Ai - Ai * A).norm() < tolerance                       #  the Euclidean norm

True

True

infinity & NaN (Not a Number)

1./0

+infinity

+infinity

1/0

Traceback (click to the left of this block for traceback)
...
ZeroDivisionError: rational division by zero

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_112.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("MS8w"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmp3fMu04/___code___.py", line 3, in <module>
    exec compile(u'_sage_const_1 /_sage_const_0 
  File "", line 1, in <module>
    
  File "sage/rings/integer.pyx", line 2047, in sage.rings.integer.Integer.__div__ (build/cythonized/sage/rings/integer.c:14042)
ZeroDivisionError: rational division by zero

-1./0

-infinity

-infinity

log (0.)

-infinity

-infinity

log (0)

-Infinity

-Infinity

oo

+Infinity

+Infinity

show (-oo)

$\newcommand{\Bold}[1]{\mathbf{#1}}-\infty$

parent (oo); parent (1./0)

The Infinity Ring
Real Field with 53 bits of precision

The Infinity Ring
Real Field with 53 bits of precision

parent (log (0.)); parent (log (0))

Real Field with 53 bits of precision
Symbolic Ring

Real Field with 53 bits of precision
Symbolic Ring

1./0 == oo

True

True

(1./0).is_infinity()

True

True

1. + oo

+Infinity

+Infinity

1. + 1./0

+infinity

+infinity

3 * oo

+Infinity

+Infinity

oo + oo

+Infinity

+Infinity

oo * oo

+Infinity

+Infinity

oo^1000

+Infinity

+Infinity

(1./0)^1000

+infinity

+infinity

2^oo

Traceback (click to the left of this block for traceback)
...
TypeError: unsupported operand parent(s) for ^: 'The Infinity Ring' and 'The
Infinity Ring'

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_129.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Ml5vbw=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpPRg_2E/___code___.py", line 3, in <module>
    exec compile(u'_sage_const_2 **oo
  File "", line 1, in <module>
    
  File "sage/rings/integer.pyx", line 2213, in sage.rings.integer.Integer.__pow__ (build/cythonized/sage/rings/integer.c:14884)
  File "sage/structure/coerce.pyx", line 1163, in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:9859)
  File "sage/structure/element.pyx", line 2038, in sage.structure.element.Element.__pow__ (build/cythonized/sage/structure/element.c:14037)
  File "sage/structure/element.pyx", line 2079, in sage.structure.element.Element._pow_ (build/cythonized/sage/structure/element.c:14381)
TypeError: unsupported operand parent(s) for ^: 'The Infinity Ring' and 'The Infinity Ring'

2^(1./0)

+infinity

+infinity

0./0.

NaN

NaN

oo - oo

Traceback (click to the left of this block for traceback)
...
sage.rings.infinity.SignError: cannot add infinity to minus infinity

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_132.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("b28gLSBvbw=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpHCRCdg/___code___.py", line 2, in <module>
    exec compile(u'oo - oo
  File "", line 1, in <module>
    
  File "sage/structure/element.pyx", line 1359, in sage.structure.element.Element.__sub__ (build/cythonized/sage/structure/element.c:11339)
  File "sage/structure/element.pyx", line 2371, in sage.structure.element.ModuleElement._sub_ (build/cythonized/sage/structure/element.c:15388)
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 429, in _sub_
    raise SignError("cannot add infinity to minus infinity")
sage.rings.infinity.SignError: cannot add infinity to minus infinity

RR (oo) + RR (-oo)

NaN

NaN

1./0 + log (0.)

NaN

NaN

0 * oo

Traceback (click to the left of this block for traceback)
...
sage.rings.infinity.SignError: cannot multiply infinity by zero

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_135.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("MCAqIG9v"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpJKBbmc/___code___.py", line 3, in <module>
    exec compile(u'_sage_const_0  * oo
  File "", line 1, in <module>
    
  File "sage/rings/integer.pyx", line 1989, in sage.rings.integer.Integer.__mul__ (build/cythonized/sage/rings/integer.c:13725)
  File "sage/structure/coerce.pyx", line 1163, in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:9859)
  File "sage/structure/element.pyx", line 1517, in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:11977)
  File "sage/structure/element.pyx", line 2548, in sage.structure.element.RingElement._mul_ (build/cythonized/sage/structure/element.c:17481)
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 1369, in _mul_
    raise SignError("cannot multiply infinity by zero")
sage.rings.infinity.SignError: cannot multiply infinity by zero

0 * RR (oo)

NaN

NaN

oo / oo

Traceback (click to the left of this block for traceback)
...
sage.rings.infinity.SignError: cannot multiply infinity by zero

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_137.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("b28gLyBvbw=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmp8XKd5C/___code___.py", line 2, in <module>
    exec compile(u'oo / oo
  File "", line 1, in <module>
    
  File "sage/structure/element.pyx", line 1647, in sage.structure.element.Element.__div__ (build/cythonized/sage/structure/element.c:12537)
  File "sage/structure/element.pyx", line 2649, in sage.structure.element.RingElement._div_ (build/cythonized/sage/structure/element.c:18126)
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 481, in _div_
    return self * ~other
  File "sage/structure/element.pyx", line 1519, in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12012)
  File "sage/structure/coerce.pyx", line 1163, in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:9859)
  File "sage/structure/element.pyx", line 1517, in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:11977)
  File "sage/structure/element.pyx", line 2548, in sage.structure.element.RingElement._mul_ (build/cythonized/sage/structure/element.c:17481)
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 455, in _mul_
    raise SignError("cannot multiply infinity by zero")
sage.rings.infinity.SignError: cannot multiply infinity by zero

RR (oo) / RR (oo)

NaN

NaN

bool (NaN == NaN)

True

True

0. * RR (oo) == 0. * RR (oo)

False

False

0./0 == 0./0

False

False

0./0 == RR (oo) / RR (oo)

False

False

bool (0./0 == NaN)

True

True

(0./0).is_NaN()

True

True

bool (NaN != NaN)

False

False

0./0 != 0./0

True

True

bool (1. < NaN)

False

False

bool (1. > NaN)

False

False

bool (1. <= NaN)

False

False

bool (1. >= NaN)

False

False

def newton_raphson (f, df, x0, tol_f = 1e-7, max_iter = 20) :
    xk = x0
    fk = f (xk)
    step = 0
    
    while (not (abs (fk) < tol_f)) and (step < max_iter):
        dfk = df (xk) 
        xk = xk - fk/dfk
        fk = f (xk)
        print step, xk, fk
        step += 1
    
    return xk

def fp (x) : 
    return (x + 1)/sqrt((x + 1)^2 + 16) + (x - 5)/sqrt((x - 5)^2 + 1)
    
def dfp (x) : 
    x1 = x + 1
    x5 = x - 5
    r = -x1^2/(x1^2 + 16)^(3/2) - x5^2/(x5^2 + 1)^(3/2) 
    r += 1/sqrt(x1^2 + 16) + 1/sqrt(x5^2 + 1)
    return r

newton_raphson (fp, dfp, 7., max_iter = 10)

0 -8.99999999999999 -1.89188589083065
1 74.2610504415619 1.99848639858029
2 -49414.0501674566 -1.99999999651880
3 1.41943459326446e13 2.00000000000000
4 -infinity NaN
5 NaN NaN + NaN*I
6 NaN + NaN*I NaN + NaN*I
7 NaN + NaN*I NaN + NaN*I
8 NaN + NaN*I NaN + NaN*I
9 NaN + NaN*I NaN + NaN*I
NaN + NaN*I

0 -8.99999999999999 -1.89188589083065
1 74.2610504415619 1.99848639858029
2 -49414.0501674566 -1.99999999651880
3 1.41943459326446e13 2.00000000000000
4 -infinity NaN
5 NaN NaN + NaN*I
6 NaN + NaN*I NaN + NaN*I
7 NaN + NaN*I NaN + NaN*I
8 NaN + NaN*I NaN + NaN*I
9 NaN + NaN*I NaN + NaN*I
NaN + NaN*I

def newton_raphson (f, df, x0, tol_f = 1e-7, max_iter = 20) :
    xk = x0
    fk = f (xk)
    step = 0
    
    while (abs (fk) >= tol_f) and (step < max_iter):
        dfk = df (xk) 
        xk = xk - fk/dfk
        fk = f (xk)
        print step, xk, fk
        step += 1
    
    return xk

newton_raphson (fp, dfp, 7., max_iter = 10)

0 -8.99999999999999 -1.89188589083065
1 74.2610504415619 1.99848639858029
2 -49414.0501674566 -1.99999999651880
3 1.41943459326446e13 2.00000000000000
4 -infinity NaN
-infinity

0 -8.99999999999999 -1.89188589083065
1 74.2610504415619 1.99848639858029
2 -49414.0501674566 -1.99999999651880
3 1.41943459326446e13 2.00000000000000
4 -infinity NaN
-infinity

... i još dvije sitnice (ili pitanje dosljednosti):

sqrt (-1.)

1.00000000000000*I

1.00000000000000*I

sqrt (-1., extend = False)

Traceback (click to the left of this block for traceback)
...
ValueError: negative number -1.00000000000000 does not have a square root in the
real field

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_150.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("c3FydCAoLTEuLCBleHRlbmQgPSBGYWxzZSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpXavP9e/___code___.py", line 3, in <module>
    exec compile(u'sqrt (-_sage_const_1p , extend = False)
  File "", line 1, in <module>
    
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/functions/other.py", line 870, in sqrt
    return x.sqrt(*args, **kwds)
  File "sage/rings/real_mpfr.pyx", line 4129, in sage.rings.real_mpfr.RealNumber.sqrt (build/cythonized/sage/rings/real_mpfr.c:26558)
ValueError: negative number -1.00000000000000 does not have a square root in the real field

log (-1.)

3.14159265358979*I

3.14159265358979*I

log (1., extend = False)

Traceback (click to the left of this block for traceback)
...
TypeError: __call__() got an unexpected keyword argument 'extend'

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_152.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bG9nICgxLiwgZXh0ZW5kID0gRmFsc2Up"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpPLp2_c/___code___.py", line 3, in <module>
    exec compile(u'log (_sage_const_1p , extend = False)
  File "", line 1, in <module>
    
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/functions/log.py", line 436, in log
    return ln(args[0], **kwds)
  File "sage/symbolic/function.pyx", line 897, in sage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:10405)
TypeError: __call__() got an unexpected keyword argument 'extend'

PPPK_03

332 days ago by fresl