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hw2/feedback/Ryan_hw2_013124.m

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+%%%% ECE210-B Matlab Seminar, Homework 2.
+% james kit paul thomas ryan murphy, 1/31/24    
+% this is james' full name. -A. 
+		% <+.01> i needed this.
+
+% this should run in octave (7.3.0)!
+		% <3, turns out it also runs in MATLAB. love the comment,
+		% though.
+close all; clear; clc;
+
+%% Spatial Awareness
+% (1)
+A = reshape(0:63, 8, 8).';
+B = 2 .^ A;
+
+% (2)
+B_colvec = reshape(B, [], 1);
+
+% primes(64) = [2, 3, 5... 61]
+% since we're asking for the prime indices from index 1 to index 64,
+% getting all prime numbers within [0, 64] seems fitting
+B_primeinds = B_colvec(primes(length(B_colvec)));
+		% very clean, like it.
+
+% (3)
+B_primeinds_geometric_mean = prod(B_primeinds) .^ (1 / length(B_primeinds));
+
+% (4)
+% (1, :) -- Grab ALL of row 1 (aka the top row) in the matrix.
+% fliplr -- flip left to right
+A(1, :) = fliplr(A(1, :));	% :)
+
+% (5)
+% Keep all rows, keep all cols minus the end col
+A_lastcol_del = A(1:8, 1:end-1);
+		% that's one way! another is to assign to A (or a copy) with an
+		% empty matrix:
+		% A(:, end) = [];
+
+
+%% Smallest Addition
+% (1)
+N = 100;
+t = linspace(0, 6.66, N);
+
+erf_integrand = exp(-t .^ 2);
+
+figure;
+title("Plot of e^{-t^2}, (100pts, [0, 6.66])");
+xlabel("time");
+ylabel("e^{-t^2}");
+plot(t, erf_integrand);
+
+% (2)
+% derivative approximation,
+% form some dt being the distance between points on our time axis
+% form the derivative by dividing the formed erf_integrand over dt 
+dt = t(2) - t(1);
+dfdt = diff(erf_integrand) ./ dt;
+
+% padding op
+dfdt = dfdt([1 1:end]);
+
+% mean squared error between approximate taken and real erf deriv
+dfdt_anal = -2 .* t .* exp( -t .^ 2);
+msqerr_dfdt = (1 / N) .* mean((dfdt - dfdt_anal) .^ 2);
+		% <-.03> (repeated) this will underreport -- `mean()` already
+		% does a divide by the length of the array, so another divide
+		% by N is not necessary.
+
+figure; hold on;
+title('d/dt[e^{-t^2}] -- analytic vs approximate (100pts, [0, 6.66])');
+xlabel('time');
+ylabel('d/dt[-exp(t^2)]');
+
+plot(t, dfdt, 'g-');
+plot(t, dfdt_anal, 'r-');
+
+% (3)
+
+% integral approx
+% take the area under the curve between t(n) and t(n-1)
+% sum the area and scale by 2/sqrt(pi)
+
+figure; hold on;
+title('erf -- analytic vs approximate (100pts, [0, 6.66])');
+xlabel('time');
+ylabel('erf(t)');
+
+% baseline
+erf_anal = (sqrt(pi) ./ 2) .* erf(t);
+
+% cum sum
+erf_approx_cumsum = cumsum(erf_integrand) .* dt;
+msqerr_erf_cumsum = (1 / N) .* mean((erf_approx_cumsum - erf_anal) .^ 2);
+
+% cum trapz
+erf_approx_cumtrapz = cumtrapz(erf_integrand) .* dt;
+msqerr_erf_cumtrapz = (1 / N) .* mean((erf_approx_cumtrapz - erf_anal) .^ 2);
+
+%{
+	The mean-square error of the cumsum op is 5 orders of magnitude larger
+		than the result from the cumtrapz op.
+	Qualitatively, the cumsum approximation (blue line) is noticably further 
+		away than the cumtrapz approximation (red line). the cumsum also has
+		a non-negligible y intercept, which is much further than the cumtrapz
+		assumption. However, the cumtrapz is basically on top of the analytic
+		function (green line), which is especially novel considering both ops
+		used the same data set. 
+%}
+		% <+.02> love the explanatory comment. `cumtrapz` is
+		% surprisingly good at what it does!
+
+% check
+plot(t, erf_approx_cumsum, "b-");
+plot(t, erf_approx_cumtrapz, "r-");
+plot(t, erf_anal, "g-");