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zC^w_{!^GSW(=C<;#u#>37#Lu<*}@PbK1?i(EYS0UiG>NK`z%b2Fx+QhhUqs8b4yI~ zERBrO!rsKv&=5WBO)L$~G4xs*S)zxPse!QtQeG`7O3chjE#d;79%@m Ge>VX7O@7M& literal 0 HcmV?d00001 diff --git a/exercise3/exercise3.typ b/exercise3/exercise3.typ new file mode 100644 index 0000000..9dc09a2 --- /dev/null +++ b/exercise3/exercise3.typ @@ -0,0 +1,199 @@ +#import "@preview/cetz:0.3.2"; +#import "@preview/cetz-plot:0.1.1": plot +#import "@preview/physica:0.9.4": * +#import "@preview/plotsy-3d:0.1.0": plot-3d-parametric-surface +#import "@preview/fletcher:0.5.4" as fletcher: diagram, edge, node + +#set page(paper: "a4", margin: (x: 2.6cm, y: 2.8cm), numbering: "1 : 1") +#set par(justify: true, leading: 0.52em) + +#let FONT_SIZE = 18pt; +#set text(font: "FreeSerif", size: FONT_SIZE, lang: "us") +#show math.equation: set text(font: "Euler Math", size: (FONT_SIZE * 1.0), lang: "en") + +#set heading(numbering: none) +#show heading.where(level: 1): it => { + rect(inset: FONT_SIZE / 2)[#it] +} + +#align(center)[ + #text(size: FONT_SIZE * 2, weight: "bold")[#underline[exercise 3]] +] + +these are my solutions to the third exercise set of TMA4135. + +there should be a python source code file attached to this deliverable. +this file can be used to generate the images shown in problem 2. + +this document was created using +#link("https://typst.app/")[#text(blue.darken(5%))[typst]]. + +#v(42pt) + +#outline(title: none) + +#v(42pt) + +#pagebreak() + += problem 1 + +== a) & b) + +using the trapezoidal rule we can estimate +$ + (1) quad I_[0,1][exp] := integral_0^1 e^x dd(x), wide + (2) quad I_[0,1][ln] := integral_0^1 ln(x) dd(x) +$ + +let us solve both analytically first, starting with $(1)$ +$ + integral_0^1 e^x dd(x) = evaluated(e^x)_0^1 = e - 1 +$ +then $(2)$ +$ + integral_0^1 ln(x) dd(x) & = [x ln(x) - 1/2 dot x^2 / x - 1/6 dot x^3 / x^2 + - 1/12 dot x^4 / x^3 - dots.c]_0^1 \ + & = -underbracket((1/2 + 1/6 + 1/12 + dots.c), 1) + - cancel(lim_(t -> 0) t ln(t)) \ + & = -1, +$ +because with $p(x) = x^2 + 3 x + 2$ we get +$ + 2 + 6 + 12 + dots.c = p(0) + p(1) + dots.c +$ +so we can compute the telescope sum +$ + sum_(k=0)^oo 1/p(k) & = sum_(k=0)^oo 1/(k^2 + 3 k + 2) \ + & = sum_(k=0)^oo 1/((k+1)(k+2)) \ + & = sum_(k=0)^oo (1/(k+1) - 1/(k+2)) \ + & = 1 - cancel(1/2 + 1/2) - cancel(1/3 + 1/3) + dots.c \ + & = 1. +$ + +now we know what to expect from our estimates. + +recall the trapezoidal rule +$ + // I_[a, b][f] approx T_[a, b][f] := (b-a)/2 (f(b) - f(a)) + I_[a, b][f] approx T_[a, b][f] := (b-a)/2 sum_(i=0)^(n-1) (f(x_(i+1)) - f(x_i)) +$ + +in our case $a = 0$ and $b = 1$. for $(1) med f(x) = e^x$ we obtain +$ + T_[0, 1][exp] = 1/2 (e - 1), +$ +which yields an error of +$ + e - 1 - 1/2 (e - 1) = 1/2 (e - 1). +$ + +for $(2) med f(x) = ln(x)$ we obtain +$ + T_[0, 1][ln] = 1/2 (lim_(t->0) ln(t) - ln(1)) = 1/2 (-oo - 0) = -oo +$ +which yields an error of $oo$. + +thus we can see that using the trapezoidal rule for $ln(x)$ diverges and proves +to be unwieldy. for $exp(x)$ we can see that it was off by half the actual +value. + + +== c) + +recall the upper bound of the error for the trapezoidal rule +$ + abs(E_[a,b][f]) <= (b-a)^3 / 12 dot max_(xi in [a,b]) abs(f''(xi)) +$ +which is for $f = exp$ +$ + E_[0, 1][exp] = e/12 approx #{ calc.round(calc.e / 12, digits: 3) } +$ +as opposed to our previous $1/2 (e - 1) += #{ calc.round((calc.e - 1) / 2, digits: 3) }$. + + + +== e) + +telescope sum ish + +overlapping edges + + += problem 3 + +== a) + +to find an orthogonal basis $p_j (x)$ from the given canonical basis +$ + #let phi = $phi.alt$ + phi_0(x) equiv 1, med + phi_1(x) = x, med + phi_2(x) = x^2, med + phi_3(x) = x^3, +$ +we can use the gram-schmidt process. + +recall that +$ + p_j (x) = phi_j (x) - sum_(k=0)^(j-1) {(integral_0^1 phi_j (x) p_k (x) dd(x)) + /(integral_0^1 [p_k (x)]^2 dd(x))} p_k (x), +$ +and notice that $p_0(x) equiv 1$. + +thus we obtain +$ + p_1(x) & = phi_1(x) - (integral_0^1 phi_1(x) p_0(x) dd(x)) + / (integral_0^1 [p_0(x)]^2 dd(x)) p_0 (x) \ + & = x - (integral_0^1 x dd(x))/(integral_0^1 dd(x)) + = x - 1/2, \ + p_2(x) & = phi_2(x) - (integral_0^1 phi_2(x) p_1(x) dd(x)) + / (integral_0^1 [p_1(x)]^2 dd(x)) p_1 (x) \ + & - (integral_0^1 phi_2(x) p_0(x) dd(x)) + / (integral_0^1 [p_0(x)]^2 dd(x)) p_0 (x) \ + & = x^2 - (integral_0^1 x^2 (x - 1/2) dd(x)) + / (integral_0^1 [x-1/2]^2 dd(x)) (x - 1/2) \ + & - (integral_0^1 x^2 dd(x)) / (integral_0^1 dd(x)) + = x^2 - x + 1/6 \ + p_3(x) & = limits(dots)^#[magic] = x^3 - 3/2 x^2 + 3/5 x - 1/20 +$ + +== b) + +we perform polynomial division to find the remaining two roots and obtain +$ + p_3 (x) : (x - 1/2) = x^2 - x + 1/10 \ + => x = 1 / 2 plus.minus 1/2 sqrt(3/5) +$ + +so +$ + x_0 & = 1/2 - 1/2 sqrt(3/5), \ + x_1 & = 1/2, \ + x_2 & = 1/2 + 1/2 sqrt(3/5). +$ + + +== c) + +let +$ + w_i = integral_0^1 ell_i (x) dd(x) +$ +with +$ + ell_k (x) = product_(i = 0\ i != k)^(n - 1) (x - x_i) / (x_k - x_i) +$ +where $n - 1 = 2$ since $k = 0, 1, 2$. + +computing the cardinal polynomials, we obtain +$ + ell_0(x) & = (x - x_1) / (x_0 - x_1) dot (x - x_2) / (x_0 - x_2) + // = (x^2 - x_1 x - x_2 x + x_1 x_2) / (x_0^2 - x_1 x_0 - x_2 x_0 + x_1 x_2) + = (1 - 2 x) / sqrt(3/5) dot (x - 1/2 + 1/2 sqrt(3/5)) / sqrt(3/5) \ + & = (5 (x - 1/2 + 1/2 sqrt(3/5) - 2x^2 - x + x sqrt(3/5)))/3 \ + & = 5 / 3 (-2 x^2 + sqrt(3/5) x + 1/2 sqrt(3/5) - 1/2) \ + ell_1(x) & = (x - x_0) / (x_1 - x_0) dot (x - x_2) / (x_1 - x_2) + = (2 x - 1 + sqrt(3/5))/(sqrt(3/5)) dot (2 x - 1 + sqrt(3/5)) / sqrt(3/5) \ +$