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+#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] +} +// #show heading.where(level: 2): it => [ +// #set align(left) +// #set text(size: FONT_SIZE * 1.1, weight: "semibold") +// #(it.body) +// ] + +#align(center)[ + #text(size: FONT_SIZE * 2, weight: "bold")[#underline[exercise 1]] +] + +these are my solutions to the second 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) + + += problem 1 + +define the _cardinal functions_ as +$ + ell_i(x) & := product_(j=0\ j!=i)^n (x - x_j) / (x_i - x_j) \ + & = (x - x_0)/(x_i - x_0) dot dots.c dot + (x - x_(i-1))/(x_i - x_(i-1)) dot (x - x_(i+1))/(x_i - x_(i+1)) + dot dots.c dot (x - x_n)/(x_i - x_n), +$ +for $i = 0, ..., n$. + + +== a) + +given $x_0 = -3, x_1 = -1, #[and] x_2 = 5$, then +#table( + $ + l_0 & = product_(j=0\ j!=i)^n (x - x_j)/(x_0 - x_j) \ + & = (x - x_1)/(x_0 - x_1) dot (x - x_2)/(x_0 - x_2) \ + & = (x + 1)/(-2) dot (x - 5)/(-8) \ + & = ((x + 1)(x - 5))/16 \ + & = (x^2 - 4x - 5)/16 + $, + $ + l_1 & = product_(j=0\ j!=i)^n (x - x_j)/(x_1 - x_j) \ + & = (x - x_0)/(x_1 - x_0) dot (x - x_2)/(x_1 - x_2) \ + & = (x + 3)/(2) dot (x - 5)/(-6) \ + & = ((x + 3)(x - 5))/(-12) \ + & = (-x^2 + 2x + 15)/12 + $, + $ + l_2 & = product_(j=0\ j!=i)^n (x - x_j)/(x_2 - x_j) \ + & = (x - x_0)/(x_2 - x_0) dot (x - x_1)/(x_2 - x_1) \ + & = (x + 3)/(8) dot (x + 1)/(6) \ + & = ((x + 3)(x + 1))/48 \ + & = (x^2 + 4x + 3)/48 + $, + + column-gutter: 20%, + stroke: none, + columns: 3, +) + + +== b) + +we can easily verify these cardinal functions +$ + l_0(x_0) & = ((-3)^2 - 4(-3) - 5)/16 = (9 + 12 - 5)/16 = 1 \ + l_0(x_1) & = ((-1)^2 - 4(-1) - 5)/16 = (1 + 4 - 5)/16 = 0 \ + l_0(x_2) & = (5^2 - 4 dot 5 - 5)/16 = (25 - 20 - 5)/16 = 0 \ + \ + \ + l_1(x_0) & = (-(-3)^2 + 2(-3) + 15)/12 = (-9 - 6 + 15)/12 = 0 \ + l_1(x_1) & = (-(-1)^2 + 2(-1) + 15)/12 = (-1 -2 + 15)/12 = 1 \ + l_1(x_2) & = (-5^2 + 2 dot 5 + 15)/12 = (-25 + 10 + 15)/12 = 0 \ + \ + \ + l_2(x_0) & = ((-3)^2 + 4(-3) + 3)/48 = (9 - 12 + 3)/48 = 0 \ + l_2(x_1) & = ((-1)^2 + 4(-1) + 3)/48 = (1 -4 + 3)/48 = 0 \ + l_2(x_2) & = (5^2 + 4 dot 5 + 3)/48 = (25 + 20 + 3)/48 = 1 +$ + +thus we see that they follow the desired pattern. + + +== c) + +assume we have $n + 1$ data points, then +$ + p_n(x) := sum_(i=0)^n y_i ell_i(x) +$ +must be the (unique) interpolation polynomial for points $(x_i, y_i)$. + +to show this, we first must convince ourselves that $deg(ell_i (x)) = n$. +recall the definition +$ + ell_i(x) & := product_(j=0\ j!=i)^n (x - x_j) / (x_i - x_j) \ + => ell_i(x) & = product_(j=0\ j!=i)^n f(x) +$ +where +$ + f(x) = (x - x_j) / (x_i - x_j) \ + => deg(f(x)) = 1 +$ + +the product contains exactly $n$ factors, because we iterate over the $n + 1$ +data points, minus the one for $i = j$. thus, $deg(ell_i(x)) = n$. + +lastly, we must verify that the interpolation condition holds for $p_n(x)$ +$ + p_n(x_i) = y_i, quad i = 0, ..., n. +$ + +we can expand the definition to show this +$ + p_n(x_i) & = sum_(j=0)^n y_j ell_j(x_i) \ + & = sum_(j=0)^n y_j product_(k=0\ k!=j)^n (x_i - x_k) / (x_j - x_k) \ + & = sum_(j=0)^n y_j dot g(i, j) +$ +where +$ + g(i, j) := cases( + 1 & quad "if" i = j, + 0 & quad "otherwise" + ) +$ + +thus we get +$ + p_n(x_i) & = (sum_(j=0\ j!=i)^n y_j dot 0) + (sum_(j=0\ j=i)^n y_j dot 1) \ + & = (0) + (y_i) = y_i +$ + +as such, we can see that $p_n(x)$ is the interpolation polynomial of the given +data points. + + +== d) + +recall from a) that given $x_0 = -3, x_1 = -1, #[and] x_2 = 5$, then +#table( + $ + l_0 = (x^2 - 4x - 5)/16 + $, + $ + l_1 = (-x^2 + 2x + 15)/12 + $, + $ + l_2 = (x^2 + 4x + 3)/48 + $, + + column-gutter: 20%, + stroke: none, + columns: 3, +) + +let $y_0 = 8, y_1 = -4, #[and] y_2 = 12$. then using our proven formula from c), +we can compute the interpolation polynomial +$ + p(x) & = sum_(j=0)^2 y_j ell_j(x) \ + & = y_0 ell_0(x) + y_1 ell_1(x) + y_2 ell_2(x) \ + & = (x^2 - 4x - 5)/2 + (x^2 - 2x - 15)/3 + (x^2 + 4x + 3)/4 \ + & = (6x^2 - 24x - 30 + 4x^2 - 8x - 60 + 3x^2 + 12x + 9)/12 \ + & = (13x^2 - 20x - 81)/12. +$ + + += problem 2 + +== a) + +#image("x.png", width: 80%) + +== b) + +#image("y.png", width: 80%) + +== c) & d) + +#image("path.png", width: 80%) + +#pagebreak() + += problem 3 + +let $y = f(x) := 2 + sin(x) cos(x)$ with +#[ + #set align(center) + #set table( + stroke: (x, y) => { + if y == 0 { (bottom: luma(180)) } + if x == 0 { (right: luma(180)) } + }, + inset: 10% + 5pt, + ) + #table( + $i$, $0$, $1$, $2$, + $x_i$, $0$, $pi/6$, $pi/3$, + $y_i$, $2$, $2 + sqrt(3)/4$, $2 + sqrt(3)/4$, + columns: 4, + ) +] + +and define +$ + p(x) := a_1 + a_2 sin x + a_3 sin 2x +$ +where $p(x_i) = y_i #[for] i in {0, 1, 2} #[and] a_1, a_2, a_3 in RR$. + +== a) + +from some quick analysis, we can see that +$ + a_1 = y_1 = 2 => \ + sqrt(3)/4 = a_2/2 + sqrt(3) a_3 / 2 and \ + sqrt(3)/4 = sqrt(3) a_2 / 2 + sqrt(3) a_3 / 2 => \ + a_2/2 + sqrt(3) a_3 / 2 = sqrt(3) a_2 / 2 + sqrt(3) a_3 / 2 \ + a_2 = sqrt(3) a_2 => a_2 = 0 => \ + a_3 = 1/2 +$ + +so we get $p(x) = 2 + 1/2 sin 2x$. + +== b) + +now, we can show that $p(x) = f(x)$, i.e. the error $e(x) = 0$ + +recall that +$ + sin 2x = 2 sin x cos x +$ + +we then get +$ + p(x) & = 2 + 1/2 sin 2x \ + & = 2 + 1/2 (2 sin x cos x) \ + & = 2 + sin x cos x = f(x) +$ +as expected. + += problem 4 + +define +$ + c_0 := y_0, quad + p_0(x) := c_0 +$ +and recursively +$ + c_k & := (y_k - p_(k-1)(x_k))/(product_(i=0)^(k-1) (x_k - x_i)), \ + p_k (x) & := p_(k-1)(x) + c_k product_(i=0)^(k-1) (x-x_i), +$ +for $k = 1, ..., n$. + +== a) + +given data points +#[ + #set align(center) + #set table( + stroke: (x, y) => { + if y == 0 { (bottom: luma(180)) } + if x == 0 { (right: luma(180)) } + }, + inset: 10% + 5pt, + ) + #table( + $i$, $0$, $1$, $2$, + $x_i$, $-1$, $1$, $5$, + $y_i$, $6$, $-2$, $4$, + columns: 4, + ) +] + +compute $p_2(x)$. + +first we need +$ + p_1(x) & = p_0(x) + (y_1 - p_0(x_1))/(x_1 - x_0) dot (x - x_0) \ + & = 6 + (-2 - 6)/(1 - (-1)) dot (x - (-1)) \ + & = 6 - 4 (x + 1) = -4x + 2 +$ +then use $p_1(x)$ in +$ + p_2(x) & = p_1(x) + (y_2 - p_1(x_2))/((x_2 - x_0)(x_2 - x_1)) + dot (x - x_0)(x - x_1) \ + & = -4x + 2 + (4 - (-4)dot 4 + 2)/((5-(-1))(5-1)) + dot (x - (-1))(x - 1) \ + & = -4x + 2 + 11/12 dot (x + 1)(x - 1) \ + & = 11/12 x^2 - 4x + 13/12 +$ + +#linebreak() + +== b) + +$p_k (x)$ is the interpolation polynomial for the data points, because +#[ #set math.equation(numbering: "(1)", supplement: none) + $ + deg(p_k (x)) = k quad forall quad k in NN quad + $ + and + $ + p_k (x_i) = y_i quad forall quad i = 0, ..., k quad + $ +] + +to show @p1, we can see from the definition that the product has degree $k$, +because there are $k$ factors containing $x$. by inductive argument, we can +trivially see that $p_(k-1) (x)$ must have a degree equal to $k-1$. as such, +$ + deg(p_k (x)) = k +$ + +next, (@p2) can be shown by a splitting the problem into two cases, one of them +requiring induction, otherwise just direct proofs. observe +$ + p_k (x_k) & = p_(k-1) (x_k) + c_k product_(i=0)^(k-1) (x_k - x_i) \ + & = cancel(p_(k-1) (x_k)) + (y_k - cancel(p_(k-1) (x_k)))/cancel( + product_(i=0)^(k-1) (x_k + - x_i) + ) dot cancel(product_(i=0)^(k-1) (x_k - x_i)) \ + & = y_k. +$ +thus (@p2) holds for $x_k$. to show that it holds for $x_j$ where $j < k$ +requires that we analyze the product in the definition: +$ + product_(i=0)^(k-1) (x_j - x_i) +$ +but since $j in {0, ..., k - 1}$, the product must contain a $0$-factor for any +chosen $j$, thus canceling the entire second term of the definition, leaving us +with +$ + p_k (x_j) & = p_(k-1) (x_j). +$ +then we need to show that +$ + p_(k-1) (x_j) & = y_j quad forall quad j in {0, ..., k - 1} +$ + +let us perform an inductive argument. the hypothesis is +$ + p_0 (x_j) = y_j +$ + +this is wrong. + +== c) + +we can use the points from a) to form a scheme + +#[ + #set align(center) + #set table( + stroke: (x, y) => { + if x == 0 { (right: luma(180)) } + }, + inset: 10% + 5pt, + ) + #table( + $x_0 = -1$, $y_(0,0) = 6$, none, none, + none, none, $y_(0,1) = -4$, none, + $x_1 = 1$, $y_(1,1) = -2$, none, $y_(0, 2) = 13 slash 12$, + none, none, $y_(1,2) = 3 slash 2$, none, + $x_2 = 5$, $y_(2,2) = 4$, none, none, + columns: 4, + ) +] + +then we can form the interpolation polynomial according to the formula +$ + p_k (x) = sum_(i=0)^k y_(0, i) product_(j=0)^(i-1) (x-x_j) +$ +such that for $n + 1 = 3$ we get +$ + p_n (x) = p_2(x) & = 6 - 4(x - (-1)) + 13/12 (x - (-1))(x - 1) \ + & = 6 - 4x - 4 + 13/12 (x^2 - 1) \ + & = 13/12 x^2 - 4x + 11/12 +$ +which is _almost_ what we got in a) suggesting that i've made a slight error in +either task. diff --git a/exercise2/main.py b/exercise2/main.py new file mode 100644 index 0000000..46cf299 --- /dev/null +++ b/exercise2/main.py @@ -0,0 +1,55 @@ +from sympy import prod, cos, sin +from sympy.abc import t +from sympy.plotting import plot, plot_parametric + +t_i = [0, 0.8976, 1.7952, 2.6928, 3.5904, 4.4880, 5.3856, 6.2832] +x_i = [1, 1.5984, -0.6564, -1.6828, -0.1191, 0.2114, -0.3514, 1] +y_i = [0, 0.7818, 0.9750, 0.4339, -0.4339, -0.975, -0.7818, 0] +n = len(x_i) + + +def cardinal(x, i, data_x): + return prod([(x - data_x[j]) / (data_x[i] - data_x[j]) for j in range(n) if i != j]) + + +def lagrange_polynomial(x, k, data_x, data_y): + return sum(data_y[i] * cardinal(x, i, data_x) for i in range(k)) + + +n = len(x_i) +domain = (t, t_i[0], t_i[-1]) + +# a) +x_t = lagrange_polynomial(t, n, t_i, x_i) +plot( + x_t, + domain, + title="x(t) – interpolated", + xlabel="t", + ylabel="x", + show=False, +).save("x.png") + +# b) +y_t = lagrange_polynomial(t, n, t_i, y_i) +plot( + y_t, + domain, + title="y(t) – interpolated", + xlabel="t", + ylabel="y", + show=False, +).save("y.png") + +# c) & d) +interpolated = (x_t, y_t) +actual = (cos(t) + sin(2 * t), sin(t)) +plot_parametric( + (x_t, y_t, domain, "interpolated"), + (cos(t) + sin(2 * t), sin(t), domain, "actual"), + title="helicopter path - approx. vs actual", + xlabel="x", + ylabel="y", + legend=True, + show=False, +).save("path.png") diff --git a/exercise2/path.png b/exercise2/path.png new file mode 100644 index 0000000000000000000000000000000000000000..2cb1a91b6d6c01428252b4975c191f5bd0d387f5 GIT binary patch literal 43134 zcmeAS@N?(olHy`uVBq!ia0y~yU}|7sV0^&A#=yW}dhyN^1_lPJ64!{5;QX|b^2DN4 z2H(Vzf}H%4oXjMJvecsD%=|oKJwpRMQyqnjl9GaAD}DW3xC*`eqICVp-x<~n3=EtF z9+AZi4E%{8%(%jSc_stH@>iZNjv*Cu-rOy(3BLOO_{Z=`YQE{aUM!fxl=|K_^)#Df z>Ixt6MHUiy`?-Ey)_){-;=}5O4kayTvnhK|-K)OLq#IlE{>h(z93k1K^JXvmYa`!p zbE~?z==09KnzhC6Y@cUM3F1<05pd!NOv}iPZ4hwcP;AkN6Jlruu}(>FHh=^KQhQh& 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