From 4933d7ee758e063a970cf70c3fef58ab14e1a3fc Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 18 May 2021 12:39:15 +0200 Subject: [PATCH] add a week of exercises --- sose2021/algebra/alg5.pdf | Bin 0 -> 1571807 bytes sose2021/algebra/alg5.xopp | Bin 0 -> 1917507 bytes sose2021/funktheo/funk5.pdf | Bin 0 -> 797818 bytes sose2021/funktheo/funk5.xopp | Bin 0 -> 957749 bytes sose2021/r/w06/P06-1.R | 14 +++++ sose2021/r/w06/P06-2.R | 34 ++++++++++ sose2021/r/w06/P06-3.R | 104 +++++++++++++++++++++++++++++++ sose2021/tut-ana/praesenz-5.xopp | Bin 0 -> 1019472 bytes 8 files changed, 152 insertions(+) create mode 100644 sose2021/algebra/alg5.pdf create mode 100644 sose2021/algebra/alg5.xopp create mode 100644 sose2021/funktheo/funk5.pdf create mode 100644 sose2021/funktheo/funk5.xopp create mode 100644 sose2021/r/w06/P06-1.R create mode 100644 sose2021/r/w06/P06-2.R create mode 100644 sose2021/r/w06/P06-3.R create mode 100644 sose2021/tut-ana/praesenz-5.xopp diff --git 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+rec_series(rep(1, 4), sum, 16) +rec_series(c(1,1), function(x) 2*x[1]+x[2], 16) +rec_series(c(1,1), function(x) x[2]-x[1], 16) +rec_series(1:5, mean, 16) +rec_series(1:5, median, 16) diff --git a/sose2021/r/w06/P06-2.R b/sose2021/r/w06/P06-2.R new file mode 100644 index 0000000..9a06848 --- /dev/null +++ b/sose2021/r/w06/P06-2.R @@ -0,0 +1,34 @@ +# Josua Kugler, Christian Merten + +make_store <- function(n) { + dl <- vector(mode="list", length=n) + i <- 1 + function(item=NULL, reset=FALSE) { + if (reset) { dl <<- vector(mode="list", length=n); i <<- 1 } + if (is.null(item)) return(dl) + dl[[i]] <<- item + i <<- i+1 + if (i > n) i <<- 1 + invisible(dl) + } +} + +store3 <- make_store(3) +store2 <- make_store(2) +store3(1) +store3("asdf") +str(store3()) +store3(list()) +store3(TRUE) +store3(1:10) +str(store3()) +store3(NA, reset=TRUE) +str(store3()) +store3(1:10) +str(store3()) +store2(1) +store2(2) +store2(3) +str(store2()) +str(store3()) + diff --git a/sose2021/r/w06/P06-3.R b/sose2021/r/w06/P06-3.R new file mode 100644 index 0000000..75b2a43 --- /dev/null +++ b/sose2021/r/w06/P06-3.R @@ -0,0 +1,104 @@ +# Josua Kugler, Christian Merten + +# a) ----------------------------------------------------------------------- +midpoint <- function(f, a, b) (b-a) * f((a+b)/2) +trapezoid <- function(f, a, b) (b-a) * (f(a) + f(b))/2 + +# b) ---------------------------------------------------------------------- +nc_integrate <- function(f, lower, upper, n, rule) { + xs <- seq(lower, upper, length.out=n+1) + sum(rule(f, xs[-(n+1)], xs[-1])) +} + +# c) ------------------------------------------------------------------------ +closeable_seq <- function(from, to, len, closed=TRUE) { + if (closed) seq(from, to, length.out = len) + else seq(from, to, length.out = len+2)[-c(1, len+2)] +} + +newton_cotes <- function(coef, closed=TRUE) { + m <- length(coef) + w <- coef / sum(coef) + function(f, a, b) { + t <- mapply(closeable_seq, a, b, len=m, closed=closed) + (b-a) * colSums(w*matrix(f(t), ncol=length(a))) + } +} + +# d) ------------------------------------------------------------------------ + +# some objects for param_fun-list +sin1x <- function(x) { + y <- suppressWarnings(sin(1/x)) + y[is.na(y)] <- 0 + y +} +set.seed(0) +x <- c(0:1, runif(5)) +y <- runif(7) + +# list of functions with integral interval +param_fun <- list( + poly = list(f = function(x) x^4 - x^3 - 3*x^2 + x + 2, lower = 0, upper = 2), + sin1x = list(f = sin1x, lower = 0, upper = 1), + lin = list(f = approxfun(x, y), lower = 0, upper = 1), + spline = list(f = splinefun(x, y), lower = 0, upper = 1) +) + +# true values of integrals +true <- c( + poly = 0.4, + sin1x = 0.504067061906928, + lin = 0.472878602164825, + spline = 0.97236924451286) + +# options for creating integral-rules +param_rule <- list( + midpoint = list(coef = 1, closed=FALSE), + trapezoid = list(coef = c(1,1), closed=TRUE), + simpson = list(coef = c(1,4,1), closed=TRUE), + boole = list(coef = c(7,32,12,32,7), closed=TRUE), + open5 = list(coef = c(611, -453, 562, 562, -453, 611), closed=FALSE) +) + +# how many function evaluations are allowed +param_n <- 5*(2:20) + +library(tidyverse) + +param <- as_tibble( + expand.grid(n = param_n, + fun_name = names(param_fun), + rule_name = names(param_rule), + stringsAsFactors=FALSE)) + +nc_rules <- sapply(param_rule, do.call, what=newton_cotes) + +# run nc_integrate with given options +call_nc <- function(n, fun_name, rule_name) { + opts <- param_fun[[fun_name]] + opts$rule = nc_rules[[rule_name]] + # to make things fair: + # a rule which evaluates f at m points may only be called n/m times + opts$n = round(n / length(param_rule[[rule_name]]$coef)) + do.call(nc_integrate, opts) +} + +param %>% rowwise() %>% + mutate(value=nc_integrate(param_fun[[fun_name]]$f, + lower = param_fun[[fun_name]]$lower, + upper = param_fun[[fun_name]]$upper, + n = n, + rule = nc_rules[[rule_name]]), + true=true[[fun_name]], + error=abs(true-value)) -> res + +# plot results +plots <- lapply(names(param_fun), function(nm) + res %>% + filter(fun_name == nm) %>% + ggplot(aes(x = n, y = error, color = rule_name)) + + scale_y_log10() + + geom_line() + geom_point() + labs(title = nm) +) +gridExtra::grid.arrange(grobs = plots, nrow=2) diff --git a/sose2021/tut-ana/praesenz-5.xopp b/sose2021/tut-ana/praesenz-5.xopp new file mode 100644 index 0000000000000000000000000000000000000000..646a0e309bdba0f0fe08d375fd0d2e9e47ce5e1b GIT binary patch literal 1019472 zcmV(rK<>XEiwFP!000001GK%@vLiRsC3-(k(b4CUahg>HS2*!W6ZGDD-qCyS`Sd4O zVyibv{hFAIPm&^38z8GtKw{-e2ma@O-#6HqS6{y3cn+wow%+fM%Y zx!wKz-+%tE|J-+%<94wAF*;5L=iTv-pIz4wiTt3i{0it>)5Uu7`#$m~ed)Y7uP1+g ze{#GVT{e^L`N#kJ<20H4n4i!4)BhKZ&KBqSW%z5f+eE(}v)_MT{U_>f^S}PHKTfWT z$?eZYJ6@^Zyh3KSi_t&jMr)pG&FCtPzvIzFEJy!O&JJF&h7ZIy`C{Fx#iF-lG**qw zu7gRFn)`ikovO9#a$E67gKn+mt4_p-wy=ZzBwuW;&V^Ju(QDkA^<<{;Os2}G^nF`i zOcL#N*x61_wI#8vstv!?qVA`23NE7$77n(i0U2ciA}ng zkjbf)<4ZAh8!uO~dDdnun$hSw70cynqiV!NWv*P0RhOQp;*HoXc3InHuTt3u4Lz>L 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