using DocumenterSlate
verify_slate_bundle("giac_intro.tar.gz")

using KaimonSlate
KaimonSlate.serve_notebook("giac_intro.jl"; inactive = true)

using KaimonSlate
KaimonSlate.serve_notebook("giac_intro.jl"; inactive = true)

Giac ready — variables x, a, t, s; helpers tex / mathblock / taylor_poly loaded.

Symbolic Computation with Giac.jl

A computer algebra system, live in a Slate notebook

Giac (the engine behind Xcas) gives us exact algebra, calculus, and linear algebra. Here we drive it from Julia through Giac.jl and let Slate render every result as typeset mathematics — no floating-point roundoff anywhere below.

1. Algebra — exact, not approximate

Factoring, expanding, simplifying, and partial-fraction decomposition all return closed forms, which Slate typesets directly.

mathblock([
    "x^4 - 1 &= " * tex(factor(x^4 - 1)),
    "(x+2)^4 &= " * tex(expand((x + 2)^4)),
    "\\frac{x^2+3x+2}{x+1} &= " * tex(simplify((x^2 + 3x + 2) / (x + 1))),
    "\\frac{1}{x^2-1} &= " * tex(partfrac(1 / (x^2 - 1))),
])

$$
\begin{aligned}x^4 - 1 &= \left(x-1\right) \left(x+1\right) \left(x^{2}+1\right) \\[6pt] (x+2)^4 &= x^{4}+8 x^{3}+24 x^{2}+32 x+16 \\[6pt] \frac{x^2+3x+2}{x+1} &= x+2 \\[6pt] \frac{1}{x^2-1} &= \frac{1}{2 \left(x-1\right)}-\frac{1}{2 \left(x+1\right)}\end{aligned}
$$

Equation solving returns exact roots — rational, irrational, and complex.

# Exact roots, including irrational and complex ones
mathblock([
    "x^2 - 5x + 6 = 0 &\\;\\Rightarrow\\; x \\in " * tex(solve(x^2 - 5x + 6, x)),
    "x^2 - 2 = 0 &\\;\\Rightarrow\\; x \\in " * tex(solve(x^2 - 2, x)),
    "x^2 + 1 = 0 &\\;\\Rightarrow\\; x \\in " * tex(solve(x^2 + 1, x)),
    "x^3 - x = 0 &\\;\\Rightarrow\\; x \\in " * tex(solve(x^3 - x, x)),
])

$$
\begin{aligned}x^2 - 5x + 6 = 0 &\;\Rightarrow\; x \in \left[2,3\right] \\[6pt] x^2 - 2 = 0 &\;\Rightarrow\; x \in \left[-\sqrt{2},\sqrt{2}\right] \\[6pt] x^2 + 1 = 0 &\;\Rightarrow\; x \in \left[\right] \\[6pt] x^3 - x = 0 &\;\Rightarrow\; x \in \left[-1,0,1\right]\end{aligned}
$$

2. Calculus — derivatives, integrals, limits

The engine differentiates and integrates in closed form and evaluates limits, including improper ones.

f = x * sin(x)
mathblock([
    "\\frac{d}{dx}\\,x\\sin x &= " * tex(diff(f, x)),
    "\\int x\\,e^{x}\\,dx &= " * tex(integrate(x * exp(x), x)),
    "\\int_{0}^{\\pi}\\sin x\\,dx &= " * tex(integrate(sin(x), x, 0, Giac.giac_eval("pi"))),
    "\\lim_{x\\to 0}\\frac{\\sin x}{x} &= " * tex(limit(sin(x) / x, x, 0)),
    "\\lim_{x\\to\\infty}\\left(1+\\tfrac{1}{x}\\right)^{x} &= " * tex(limit((1 + 1/x)^x, x, Inf)),
])

$$
\begin{aligned}\frac{d}{dx}\,x\sin x &= \sin x+x \cos x \\[6pt] \int x\,e^{x}\,dx &= x \mathrm{e}^{x}-\mathrm{e}^{x} \\[6pt] \int_{0}^{\pi}\sin x\,dx &= 2 \\[6pt] \lim_{x\to 0}\frac{\sin x}{x} &= 1 \\[6pt] \lim_{x\to\infty}\left(1+\tfrac{1}{x}\right)^{x} &= \mathrm{e}\end{aligned}
$$

3. Interactive: Taylor series, live

Pick a function and a truncation order. Giac computes the Taylor polynomial about $x = 0$ symbolically; we compile it to a Julia function with build_function and plot it against the exact curve. Push the order up and watch the polynomial hug the true function — then notice how $\arctan x$ still diverges past its radius of convergence $|x| = 1$, no matter how many terms you add.

@bind fkey Select(["sin(x)" => "sin x", "cos(x)" => "cos x",
                   "exp(x)" => "exp x", "atan(x)" => "arctan x"];
                  label="f(x)")
@bind ord Slider(1:2:15; default=5, label="order n")

fexpr  = Giac.giac_eval(fkey.value)
poly   = taylor_poly(fexpr, ord)
approx = build_function(poly, x)
truef  = build_function(fexpr, x)

xs   = collect(range(-3, 3; length=241))
ytru = truef.(xs)
yapp = approx.(xs)

# keep the view framed on the true curve even when the series runs off
ymax = maximum(abs, ytru) * 1.6
yapp = clamp.(yapp, -ymax, ymax)

echart(series(:line, xs, ytru; name="f(x) = $(fkey.value)", smooth=true, symbol="none"),
       series(:line, xs, yapp; name="Taylor, order $ord", smooth=true, symbol="none",
              lineStyle=(type=:dashed, width=2));
       legend=true, title="Taylor approximation about x = 0",
       yAxis=(min=round(-ymax; digits=2), max=round(ymax; digits=2)),
       height=420)

KaimonSlate.ReportEngine.EChart(Dict{String, Any}("xAxis" => Dict{String, Any}("type" => "value"), "animationDurationUpdate" => 300, "__size" => Dict{String, Any}("height" => 420), "series" => Dict{String, Any}[Dict("name" => "f(x) = sin(x)", "data" => [[-3.0, -0.14112000806], [-2.975, -0.165823144744], [-2.95, -0.190422647361], [-2.925, -0.214903142022], [-2.9, -0.239249329214], [-2.875, -0.263445993363], [-2.85, -0.287478012343], [-2.825, -0.311330366922], [-2.8, -0.334988150156], [-2.775, -0.358436576701], [-2.75, -0.381660992052], [-2.725, -0.404646881708], [-2.7, -0.427379880234], [-2.675, -0.449845780247], [-2.65, -0.47203054129], [-2.625, -0.49392029861], [-2.6, -0.515501371821], [-2.575, -0.536760273456], [-2.55, -0.557683717391], [-2.525, -0.578258627157], [-2.5, -0.598472144104], [-2.475, -0.618311635442], [-2.45, -0.637764702135], [-2.425, -0.656819186648], [-2.4, -0.675463180551], [-2.375, -0.693685031953], [-2.35, -0.711473352791], [-2.325, -0.728817025942], [-2.3, 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0.684940757813], [2.575, 0.672776427653], [2.6, 0.660781333333], [2.625, 0.648992156982], [2.65, 0.637446596354], [2.675, 0.626183374593], [2.7, 0.61524225], [2.725, 0.604664025798], [2.75, 0.594490559896], [2.775, 0.584764774658], [2.8, 0.575530666667], [2.825, 0.566833316488], [2.85, 0.558718898438], [2.875, 0.551234690348], [2.9, 0.544429083333], [2.925, 0.538351591553], [2.95, 0.533052861979], [2.975, 0.528584684163], [3.0, 0.525]], "smooth" => true, "symbol" => "none", "type" => "line", "lineStyle" => Dict{String, Any}("width" => 2, "type" => "dashed"))], "backgroundColor" => "transparent", "yAxis" => Dict{String, Any}("max" => 1.6, "min" => -1.6), "tooltip" => Dict{String, Any}("trigger" => "axis"), "grid" => Dict{String, Any}("top" => 72, "containLabel" => true), "textStyle" => Dict{String, Any}("fontFamily" => "inherit"), "title" => Dict{String, Any}("text" => "Taylor approximation about x = 0"), "legend" => Dict{String, Any}("top" => 30)))

mathblock(["f(x) = $(fkey.value) &\\approx " * tex(poly)])

$$
\begin{aligned}f(x) = sin(x) &\approx x-\frac{x^{3}}{6}+\frac{x^{5}}{120}\end{aligned}
$$

4. Laplace transforms

Giac transforms between the time domain $t$ and the complex frequency domain $s$ exactly. laplace carries $f(t)$ to $F(s)$; ilaplace inverts it. This is the workhorse of linear-systems and control theory — and a round trip should return where it started.

# Forward transforms:  f(t)  ⟶  F(s) = ∫₀^∞ f(t) e^{-st} dt
mathblock([
    "\\mathcal{L}\\{1\\} &= " * tex(laplace(Giac.giac_eval("1"), t, s)),
    "\\mathcal{L}\\{t^{2}\\} &= " * tex(laplace(t^2, t, s)),
    "\\mathcal{L}\\{e^{-2t}\\sin 3t\\} &= " * tex(laplace(exp(-2t) * sin(3t), t, s)),
    "\\mathcal{L}\\{t\\,e^{-t}\\} &= " * tex(laplace(t * exp(-t), t, s)),
])

$$
\begin{aligned}\mathcal{L}\{1\} &= \frac1{s} \\[6pt] \mathcal{L}\{t^{2}\} &= \frac{2}{s^{3}} \\[6pt] \mathcal{L}\{e^{-2t}\sin 3t\} &= \frac{3}{s^{2}+4 s+13} \\[6pt] \mathcal{L}\{t\,e^{-t}\} &= \frac1{s^{2}+2 s+1}\end{aligned}
$$

# Inverse transforms:  F(s)  ⟶  f(t).  A partial-fraction split makes the
# underlying second-order system's response transparent.
F = (s + 3) / (s^2 + 2s + 5)
mathblock([
    "F(s) &= " * tex(F),
    "\\mathcal{L}^{-1}\\!\\left\\{\\tfrac{1}{s^{2}+4}\\right\\} &= " *
        tex(ilaplace(1 / (s^2 + 4), s, t)),
    "\\mathcal{L}^{-1}\\{F(s)\\} &= " * tex(ilaplace(F, s, t)),
])

$$
\begin{aligned}F(s) &= \frac{s+3}{s^{2}+2 s+5} \\[6pt] \mathcal{L}^{-1}\!\left\{\tfrac{1}{s^{2}+4}\right\} &= \frac{\sin \left(2 t\right)}{2} \\[6pt] \mathcal{L}^{-1}\{F(s)\} &= \cos \left(2 t\right) \mathrm{e}^{-t}+\sin \left(2 t\right) \mathrm{e}^{-t}\end{aligned}
$$

# Round trip: transform, invert, and simplify back to the original signal.
g       = exp(-t) * cos(2t)
G       = laplace(g, t, s)
back    = simplify(ilaplace(G, s, t))
mathblock([
    "g(t) &= " * tex(g),
    "\\mathcal{L}\\{g\\} = G(s) &= " * tex(G),
    "\\mathcal{L}^{-1}\\{G\\} &= " * tex(back),
])

$$
\begin{aligned}g(t) &= \mathrm{e}^{-t} \cos \left(2 t\right) \\[6pt] \mathcal{L}\{g\} = G(s) &= \frac{s+1}{s^{2}+2 s+5} \\[6pt] \mathcal{L}^{-1}\{G\} &= \cos \left(2 t\right) \mathrm{e}^{-t}\end{aligned}
$$

5. A second-order system, end to end

The canonical control-theory system

\[H(s) = \frac{\omega_n^{2}}{s^{2} + 2\zeta\omega_n s + \omega_n^{2}}\]

has behaviour governed entirely by its damping ratio $\zeta$ and natural frequency $\omega_n$. Giac inverts $H(s)/s$ symbolically to get the exact step response $y(t)$ — no numerical ODE solver. Drag the sliders: the closed form, the pole locations in the complex $s$-plane, and the time response all update together. Watch the poles split into a complex-conjugate pair as $\zeta$ drops below $1$ and the response begins to ring.

@bind ζ  Slider(0.1, 2.0, 0.3; step=0.05, label="damping ζ")
@bind ωn Slider(0.5, 4.0, 2.0; step=0.1, label="natural freq ωₙ")

# Giac inverts the step response Y(s) = H(s)/s to y(t), exactly.
ystep = Giac.giac_eval(
    "ilaplace($(ωn)^2/(s*(s^2+2*$(ζ)*$(ωn)*s+$(ωn)^2)), s, t)")
yfun  = build_function(ystep, t)

# Poles = roots of s² + 2ζωₙs + ωₙ²
disc  = complex(ζ^2 - 1)
poles = [-ζ*ωn + ωn*sqrt(disc), -ζ*ωn - ωn*sqrt(disc)]
regime = ζ < 1 ? "under-damped (ringing)" : ζ ≈ 1 ? "critically damped" : "over-damped"

mathblock(["y(t) &= " * tex(Giac.Commands.simplify(ystep))])

$$
\begin{aligned}y(t) &= \frac{\sqrt{91} \mathrm{e}^{\frac{3}{5} t} \tan ^{2}\left(\frac{1}{10} \sqrt{91} t\right)+\sqrt{91} \mathrm{e}^{\frac{3}{5} t}+\sqrt{91} \tan ^{2}\left(\frac{1}{10} \sqrt{91} t\right)-\sqrt{91}-6 \tan \left(\frac{1}{10} \sqrt{91} t\right)}{\sqrt{91} \mathrm{e}^{\frac{3}{5} t} \tan ^{2}\left(\frac{1}{10} \sqrt{91} t\right)+\sqrt{91} \mathrm{e}^{\frac{3}{5} t}}\end{aligned}
$$

# Pole map in the complex s-plane. Poles in the left half-plane ⇒ stable;
# imaginary part ⇒ oscillation.
R  = ωn * 1.3
px = real.(poles); py = imag.(poles)

echart(series(:scatter, px, py; name="poles", symbol="path://M0,-1 L0,1 M-1,0 L1,0",
              symbolSize=18, itemStyle=(color="#f56c6c",));
       title="Poles in the s-plane — $regime",
       xAxis=(name="Re(s)", min=-R, max=R, axisLine=(onZero=true,)),
       yAxis=(name="Im(s)", min=-R, max=R, axisLine=(onZero=true,)),
       grid=(left=60, right=30, top=50, bottom=50),
       tooltip=(formatter="Re {@[0]}, Im {@[1]}",),
       height=360)

KaimonSlate.ReportEngine.EChart(Dict{String, Any}("xAxis" => Dict{String, Any}("name" => "Re(s)", "max" => 2.6, "min" => -2.6, "axisLine" => Dict{String, Any}("onZero" => true)), "animationDurationUpdate" => 300, "__size" => Dict{String, Any}("height" => 360), "series" => Dict{String, Any}[Dict("name" => "poles", "symbolSize" => 18, "itemStyle" => Dict{String, Any}("color" => "#f56c6c"), "data" => [[-0.6, 1.9078784028338913], [-0.6, -1.9078784028338913]], "symbol" => "path://M0,-1 L0,1 M-1,0 L1,0", "type" => "scatter")], "backgroundColor" => "transparent", "yAxis" => Dict{String, Any}("name" => "Im(s)", "max" => 2.6, "min" => -2.6, "axisLine" => Dict{String, Any}("onZero" => true)), "tooltip" => Dict{String, Any}("formatter" => "Re {@[0]}, Im {@[1]}"), "grid" => Dict{String, Any}("left" => 60, "bottom" => 50, "right" => 30, "top" => 50), "textStyle" => Dict{String, Any}("fontFamily" => "inherit"), "title" => Dict{String, Any}("text" => "Poles in the s-plane — under-damped (ringing)")))

# Exact step response y(t), sampled from Giac's symbolic solution.
tmax = clamp(8 / (ζ * ωn), 4, 40)
ts   = collect(range(0, tmax; length=400))
ys   = yfun.(ts)

echart(:line, ts, ys; title="Step response y(t)  —  ζ = $ζ, ωₙ = $ωn",
       smooth=true, symbol="none", areaStyle=(opacity=0.12,),
       xAxis=(name="t",), yAxis=(name="y",),
       markLine=(silent=true, symbol="none",
                 data=[(yAxis=1, lineStyle=(type=:dashed, color="#888"))]),
       height=360)

KaimonSlate.ReportEngine.EChart(Dict{String, Any}("xAxis" => Dict{String, Any}("name" => "t"), "yAxis" => Dict{String, Any}("name" => "y"), "animationDurationUpdate" => 300, "textStyle" => Dict{String, Any}("fontFamily" => "inherit"), "tooltip" => Dict{String, Any}("trigger" => "axis"), "title" => Dict{String, Any}("text" => "Step response y(t)  —  ζ = 0.3, ωₙ = 2.0"), "__size" => Dict{String, Any}("height" => 360), "series" => Dict{String, Any}[Dict("areaStyle" => Dict{String, Any}("opacity" => 0.12), "data" => [[0.0, 0.0], [0.03341687552213868, 0.00220300106147], [0.06683375104427736, 0.00868651416391], [0.10025062656641603, 0.0192539090551], [0.1336675020885547, 0.0336984102859], [0.1670843776106934, 0.0518044009902], [0.20050125313283207, 0.0733487141135], [0.23391812865497075, 0.0981019059336], [0.2673350041771094, 0.125829506994], [0.3007518796992481, 0.156293245861], [0.3341687552213868, 0.189252241426], [0.36758563074352546, 0.224464159755], [0.40100250626566414, 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0.981398298107], [6.750208855472013, 0.98173919295], [6.783625730994152, 0.982146611215], [6.81704260651629, 0.982616154909], [6.8504594820384295, 0.983143327002], [6.883876357560568, 0.983723555001], [6.917293233082707, 0.984352213934], [6.950710108604845, 0.98502464865], [6.984126984126984, 0.985736195385], [7.017543859649122, 0.986482202506], [7.0509607351712615, 0.987258050399], [7.0843776106934, 0.988059170426], [7.117794486215539, 0.988881062919], [7.151211361737677, 0.989719314162], [7.184628237259816, 0.990569612329], [7.2180451127819545, 0.991427762354], [7.251461988304094, 0.992289699699], [7.284878863826232, 0.993151503014], [7.318295739348371, 0.994009405673], [7.351712614870509, 0.994859806175], [7.385129490392648, 0.995699277418], [7.4185463659147866, 0.996524574845], [7.451963241436926, 0.997332643465], [7.485380116959064, 0.998120623778], [7.518796992481203, 0.998885856601], [7.552213868003341, 0.999625886836], [7.58563074352548, 1.00033846619], [7.619047619047619, 1.0010215549], [7.652464494569758, 1.00167332244], [7.685881370091896, 1.00229214732], [7.719298245614035, 1.00287661598], [7.752715121136173, 1.00342552079], [7.7861319966583125, 1.00393785722], [7.819548872180451, 1.0044128203], [7.85296574770259, 1.00484980027], [7.886382623224728, 1.00524837757], [7.919799498746867, 1.00560831722], [7.953216374269005, 1.00592956259], [7.9866332497911445, 1.0062122286], [8.020050125313283, 1.00645659455], [8.053467000835422, 1.00666309635], [8.086883876357561, 1.00683231852], [8.1203007518797, 1.00696498577], [8.153717627401837, 1.0070619543], [8.187134502923977, 1.00712420291], [8.220551378446116, 1.00715282389], [8.253968253968255, 1.00714901377], [8.287385129490392, 1.00711406398], [8.320802005012531, 1.00704935151], [8.35421888053467, 1.00695632945], [8.38763575605681, 1.0068365177], [8.421052631578947, 1.00669149367], [8.454469507101086, 1.0065228831], [8.487886382623225, 1.00633235108], [8.521303258145364, 1.00612159318], [8.554720133667502, 1.00589232685], [8.58813700918964, 1.00564628304], [8.62155388471178, 1.00538519808], [8.654970760233919, 1.00511080591], [8.688387635756056, 1.00482483054], [8.721804511278195, 1.0045289789], [8.755221386800335, 1.00422493407], [8.788638262322474, 1.00391434882], [8.822055137844611, 1.00359883959], [8.85547201336675, 1.00327998084], [8.88888888888889, 1.00295929978], [8.922305764411028, 1.00263827158], [8.955722639933166, 1.00231831495], [8.989139515455305, 1.00200078814], [9.022556390977444, 1.00168698537], [9.055973266499583, 1.0013781337], [9.08939014202172, 1.00107539029], [9.12280701754386, 1.00077984005], [9.156223893065999, 1.00049249382], [9.189640768588138, 1.00021428674], [9.223057644110275, 0.999946077236], [9.256474519632414, 0.999688646215], [9.289891395154553, 0.999442696721], [9.323308270676693, 0.999208853908], [9.35672514619883, 0.998987665373], [9.390142021720969, 0.9987796018], [9.423558897243108, 0.99858505793], [9.456975772765247, 0.998404353816], [9.490392648287385, 0.998237736356], [9.523809523809524, 0.998085381089], [9.557226399331663, 0.997947394236], [9.590643274853802, 0.99782381495], [9.62406015037594, 0.997714617792], [9.657477025898078, 0.997619715373], [9.690893901420218, 0.997538961175], [9.724310776942357, 0.997472152518], [9.757727652464494, 0.997419033649], [9.791144527986633, 0.997379298953], [9.824561403508772, 0.997352596251], [9.857978279030911, 0.997338530164], [9.891395154553049, 0.997336665552], [9.924812030075188, 0.997346530974], [9.958228905597327, 0.997367622179], [9.991645781119466, 0.997399405607], [10.025062656641603, 0.997441321872], [10.058479532163743, 0.997492789227], [10.091896407685882, 0.997553206988], [10.12531328320802, 0.99762195891], [10.158730158730158, 0.997698416493], [10.192147034252297, 0.997781942216], [10.225563909774436, 0.997871892682], [10.258980785296576, 0.997967621664], [10.292397660818713, 0.998068483045], [10.325814536340852, 0.998173833641], [10.359231411862991, 0.998283035903], [10.39264828738513, 0.998395460482], [10.426065162907268, 0.998510488667], [10.459482038429407, 0.998627514667], [10.492898913951546, 0.998745947762], [10.526315789473685, 0.998865214291], [10.559732664995822, 0.998984759496], [10.593149540517961, 0.999104049204], [10.6265664160401, 0.999222571355], [10.65998329156224, 0.99933983738], [10.693400167084377, 0.999455383409], [10.726817042606516, 0.999568771333], [10.760233918128655, 0.999679589711], [10.793650793650794, 0.99978745452], [10.827067669172932, 0.999892009757], [10.86048454469507, 0.999992927901], [10.89390142021721, 1.00008991022], [10.927318295739349, 1.00018268695], [10.960735171261486, 1.00027101735], [10.994152046783626, 1.00035468958], [11.027568922305765, 1.00043352053], [11.060985797827904, 1.00050735547], [11.094402673350041, 1.00057606762], [11.12781954887218, 1.00063955761], [11.16123642439432, 1.00069775283], [11.194653299916459, 1.00075060668], [11.228070175438596, 1.0007980978], [11.261487050960735, 1.00084022913], [11.294903926482874, 1.00087702692], [11.328320802005013, 1.00090853979], [11.36173767752715, 1.00093483754], [11.39515455304929, 1.00095601006], [11.428571428571429, 1.00097216613], [11.461988304093568, 1.00098343224], [11.495405179615705, 1.00098995128], [11.528822055137844, 1.00099188131], [11.562238930659984, 1.00098939426], [11.595655806182123, 1.00098267464], [11.62907268170426, 1.00097191823], [11.6624895572264, 1.00095733078], [11.695906432748538, 1.00093912672], [11.729323308270677, 1.00091752789], [11.762740183792815, 1.00089276225], [11.796157059314954, 1.00086506267], [11.829573934837093, 1.00083466573], [11.862990810359232, 1.00080181046], [11.89640768588137, 1.00076673732], [11.929824561403509, 1.00072968697], [11.963241436925648, 1.00069089931], [11.996658312447787, 1.00065061237], [12.030075187969924, 1.00060906142], [12.063492063492063, 1.000566478], [12.096908939014202, 1.00052308908], [12.130325814536342, 1.0004791162], 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0.99964603799], [13.065998329156225, 0.999639512039], [13.099415204678362, 0.999634820367], [13.132832080200501, 0.999631911409], [13.16624895572264, 0.999630727824], [13.19966583124478, 0.999631206973], [13.233082706766917, 0.999633281402], [13.266499582289056, 0.999636879325], [13.299916457811195, 0.999641925114], [13.333333333333334, 0.999648339771]], "smooth" => true, "symbol" => "none", "markLine" => Dict{String, Any}("data" => Any[Dict{String, Any}("yAxis" => 1, "lineStyle" => Dict{String, Any}("color" => "#888", "type" => "dashed"))], "symbol" => "none", "silent" => true), "type" => "line")], "backgroundColor" => "transparent"))

6. Symbolic linear algebra

Matrices with symbolic entries carry through determinants, inverses, and eigenvalues as exact expressions in the parameter $a$.

M   = GiacMatrix(Giac.giac_eval("[[a,1],[1,a]]"))
det = Giac.invoke_cmd(:det, M)
inv = Giac.invoke_cmd(:inverse, M)
eig = Giac.invoke_cmd(:eigenvalues, M)
mathblock([
    "A &= " * tex(M),
    "\\det A &= " * tex(det),
    "A^{-1} &= " * tex(inv),
    "\\operatorname{eig}(A) &= " * tex(eig),
])

$$
\begin{aligned}A &= \left[\begin{array}{cc}a&1\\1&a\end{array}\right] \\[6pt] \det A &= a^{2}-1 \\[6pt] A^{-1} &= \left[\begin{array}{cc}\frac{a}{a^{2}-1}&-\frac1{a^{2}-1}\\-\frac1{a^{2}-1}&\frac{a}{a^{2}-1}\end{array}\right] \\[6pt] \operatorname{eig}(A) &= a+1,a-1\end{aligned}
$$

# Giac symbolic output embedded inline in markdown prose.
# `tex(e)` renders any Giac expression to a LaTeX string; drop it into a
# Markdown.parse string and it typesets alongside ordinary text.
let
    fac  = factor(x^4 - 1)
    dsol = solve(x^2 - 5x + 6, x)
    ig   = integrate(x * exp(x), x)

    Markdown.parse("""
    We can weave Giac's **symbolic output** straight into markdown prose.

    - The quartic factors as \$x^4 - 1 = $(tex(fac))\$.
    - Solving \$x^2 - 5x + 6 = 0\$ gives \$x \\in $(tex(dsol))\$.
    - And \$\\displaystyle\\int x\\,e^{x}\\,dx = $(tex(ig)) + C\$.

    The same aligned `mathblock` also renders as markdown:

    \$\$\\begin{aligned}
    x^4 - 1 &= $(tex(fac)) \\\\[6pt]
    \\int x\\,e^{x}\\,dx &= $(tex(ig)) + C
    \\end{aligned}\$\$
    """)
end

We can weave Giac's **symbolic output** straight into markdown prose.

  * The quartic factors as $x^4 - 1 = \left(x-1\right) \left(x+1\right) \left(x^{2}+1\right)$.
  * Solving $x^2 - 5x + 6 = 0$ gives $x \in \left[2,3\right]$.
  * And $\displaystyle\int x\,e^{x}\,dx = x \mathrm{e}^{x}-\mathrm{e}^{x} + C$.

The same aligned `mathblock` also renders as markdown:

$$
\begin{aligned}
x^4 - 1 &= \left(x-1\right) \left(x+1\right) \left(x^{2}+1\right) \\[6pt]
\int x\,e^{x}\,dx &= x \mathrm{e}^{x}-\mathrm{e}^{x} + C
\end{aligned}
$$

begin
    h = 10
    y = 2
end

2