result.md
In rim: Interface to 'Maxima', Enabling Symbolic Computation

title: "knitr engine test page" output: html_document

maxima.options(engine.format = "latex", engine.label = TRUE, inline.format = "inline", inline.label = FALSE)

(%i1) L: sqrt(1 - 1/R^2);$$\mathtt{(\textit{\%o}{1})}\quad \sqrt{1-\frac{1}{R^2}}$$ (%i2) assume(R > 0);$$\mathtt{(\textit{\%o}{2})}\quad \left[ R>0 \right] $$ (%i3) 'integrate(x, x, 0, L) = integrate(x, x, 0, L);$$\mathtt{(\textit{\%o}{3})}\quad \int{0}^{\sqrt{1-\frac{1}{R^2}}}{x\;dx}=\frac{R^2-1}{2\,R^2}$$

(%i4) 'L = L;$$\mathtt{(\textit{\%o}{4})}\quad L=\sqrt{1-\frac{1}{R^2}}$$ (%i5) 'integrate(x, x, 0, 'L) = integrate(x, x, 0, L);$$\mathtt{(\textit{\%o}{5})}\quad \int_{0}^{L}{x\;dx}=\frac{R^2-1}{2\,R^2}$$

This is an inline test: $L=\sqrt{1-\frac{1}{R^2}}$ .

(%i7) sqrt(3/4);$$\mathtt{(\textit{\%o}_{7})}\quad \frac{\sqrt{3}}{2}$$

(%i8) f(x) := e^(x^2)$(%i9) diff(f(x), x);$$\mathtt{(\textit{\%o}_{9})}\quad 2\,e^{x^2}\,\log e\,x$$

(%i10) %;$$\mathtt{(\textit{\%o}_{10})}\quad 2\,e^{x^2}\,\log e\,x$$

(%i11) log(%o1);$$\mathtt{(\textit{\%o}_{11})}\quad \frac{\log \left(1-\frac{1}{R^2}\right)}{2}$$

moo## $o11

eval(moo[[1]], list(R = 12))## [1] -0.003484335

Plots

(%i12) r: (exp(cos(t))-2cos(4t)-sin(t/12)^5)$(%i13) plot2d([parametric, rsin(t), rcos(t), [t,-8%pi,8%pi]]);

(%i14) plot3d(log (x^2*y^2), [x, -2, 2], [y, -2, 2],[grid, 29, 29], [palette, [gradient, red, orange, yellow, green]], color_bar, [xtics, 1], [ytics, 1], [ztics, 4], [color_bar_tics, 4]);

(%i15) example1: gr3d (title = "Controlling color range", enhanced3d = true, color = green, cbrange = [-3,10], explicit(x^2+y^2, x,-2,2,y,-2,2)) $(%i16) example2: gr3d (title = "Playing with tics in colorbox", enhanced3d = true, color = green, cbtics = {["High",10],["Medium",05],["Low",0]}, cbrange = [0, 10], explicit(x^2+y^2, x,-2,2,y,-2,2))$(%i17) example3: gr3d (title = "Logarithmic scale to colors", enhanced3d = true, color = green, logcb = true, logz = true, palette = [-15,24,-9], explicit(exp(x^2-y^2), x,-2,2,y,-2,2))$(%i18) draw( dimensions = [500,1500], example1, example2, example3);

(%i19) draw2d( dimensions = [1000, 1000], proportional_axes = xy, fill_color = sea_green, color = aquamarine, line_width = 6, ellipse(7,6,2,3,0,360));

(%i20) draw3d( dimensions = [1000, 1000], surface_hide = true, axis_3d = false, proportional_axes = xyz,

color = blue, cylindrical(z,z,-2,2,a,0,2*%pi),

color = brown, cylindrical(3,z,-2,2,az,0,%pi),

color = green, cylindrical(sqrt(25-z^2),z,-5,5,a,0,%pi));

pft <- list.files(pattern = "(?:plot|draw)(2d|3d)?-[[:print:]]{6}\.png", full.names = TRUE)if(length(pft) == 5L) { paste0("OK") } else { paste0("Error: Unexpected number of Maxima plots: ", paste0(pft, collapse = ", ")) }## [1] "OK" if(length(pft)) { if(all(as.logical(file.size(pft)))) { paste0("OK") } else { errfiles <- pft[file.size(pft) == 0] paste0("Error: Maxima plot file(s) ", paste0(errfiles, collapse = ", "), "are empty.") } }## [1] "OK"

Normal Distribution

(%i21) area(dist) := integrate(dist, x, minf, inf)$(%i22) mean(dist) := area(distx)$(%i23) EX2(dist) := area(distx^2)$(%i24) variance(dist) := EX2(dist) - mean(dist)^2$(%i25) mgf(dist) := area(dist%e^(xt))$

(%i26) normal(x) := (2%pisigma^2)^(-1/2) * exp(-(x-mu)^2/(2*sigma^2));$$\mathtt{(\textit{\%o}{26})}\quad \textit{normal}\left(x\right):=\left(2\,\pi\,\sigma^2\right)^{\frac{-1}{2}}\,\exp \left(\frac{-\left(x-\mu\right)^2}{2\,\sigma^2}\right)$$ (%i27) assume(sigma > 0)$(%i28) area(normal(x));$$\mathtt{(\textit{\%o}{28})}\quad 1$$ (%i29) mean(normal(x));$$\mathtt{(\textit{\%o}{29})}\quad \mu$$ (%i30) variance(normal(x));$$\mathtt{(\textit{\%o}{30})}\quad \frac{2^{\frac{3}{2}}\,\sqrt{\pi}\,\sigma^3+2^{\frac{3}{2}}\,\sqrt{\pi}\,\mu^2\,\sigma}{2^{\frac{3}{2}}\,\sqrt{\pi}\,\sigma}-\mu^2$$ (%i31) mgf(normal(x));$$\mathtt{(\textit{\%o}_{31})}\quad e^{\frac{\sigma^2\,t^2+2\,\mu\,t}{2}}$$

Laplace Distribution

(%i32) laplace(x) := (2*b)^-1 * exp(-abs(x - mu)/b);$$\mathtt{(\textit{\%o}_{32})}\quad \textit{laplace}\left(x\right):=\left(2\,b\right)^ {- 1 }\,\exp \left(\frac{-\left| x-\mu\right| }{b}\right)$$ (%i33) load("abs_integrate")$(%i34) assume(b > 0)$(%i35) area(laplace(x))$(%i36) mean(laplace(x))$(%i37) variance(laplace(x))$

Exponential Distribution

(%i38) expo(x) := unit_step(x) * lambda * exp(-lambda * x);$$\mathtt{(\textit{\%o}{38})}\quad \textit{expo}\left(x\right):=\textit{unit_step}\left(x\right)\,\lambda\,\exp \left(\left(-\lambda\right)\,x\right)$$ (%i39) assume(lambda > 0)$(%i40) area(expo(x));$$\mathtt{(\textit{\%o}{40})}\quad 1$$ (%i41) mean(expo(x));$$\mathtt{(\textit{\%o}{41})}\quad \frac{1}{\lambda}$$ (%i42) variance(expo(x));$$\mathtt{(\textit{\%o}{42})}\quad \frac{1}{\lambda^2}$$

Matrices

(%i43) m: matrix([0, 1, a], [1, 0, 1], [1, 1, 0]);$$\mathtt{(\textit{\%o}{43})}\quad \begin{pmatrix}0 & 1 & a \ 1 & 0 & 1 \ 1 & 1 & 0 \ \end{pmatrix}$$ (%i44) transpose(m);$$\mathtt{(\textit{\%o}{44})}\quad \begin{pmatrix}0 & 1 & 1 \ 1 & 0 & 1 \ a & 1 & 0 \ \end{pmatrix}$$ (%i45) determinant(m);$$\mathtt{(\textit{\%o}{45})}\quad a+1$$ (%i46) f: invert(m), detout;$$\mathtt{(\textit{\%o}{46})}\quad \frac{\begin{pmatrix}-1 & a & 1 \ 1 & -a & a \ 1 & 1 & -1 \ \end{pmatrix}}{a+1}$$ (%i47) m . f;$$\mathtt{(\textit{\%o}{47})}\quad \begin{pmatrix}0 & 1 & a \ 1 & 0 & 1 \ 1 & 1 & 0 \ \end{pmatrix}\cdot \left(\frac{\begin{pmatrix}-1 & a & 1 \ 1 & -a & a \ 1 & 1 & -1 \ \end{pmatrix}}{a+1}\right)$$ (%i48) expand(%);$$\mathtt{(\textit{\%o}{48})}\quad \begin{pmatrix}\frac{a}{a+1}+\frac{1}{a+1} & 0 & 0 \ 0 & \frac{a}{a+1}+\frac{1}{a+1} & 0 \ 0 & 0 & \frac{a}{a+1}+\frac{1}{a+1} \ \end{pmatrix}$$ (%i49) factor(%);$$\mathtt{(\textit{\%o}_{49})}\quad \begin{pmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{pmatrix}$$