Merge pull request #116 from fangliu-tju/main

update

quarto/alternatives/SciML.qmd

@@ -129,7 +129,7 @@ This problem can also be solved using a bracketing method. The package has both
 
 ```{julia}
 u0 = (1.0, 2.0)
-prob = NonlinearProblem(f, u0)
+prob = IntervalNonlinearProblem(f, u0)
 ```
 
 And
@@ -149,7 +149,7 @@ Incorporating parameters is readily done. For example to solve $f(x) = \cos(x) -
 f(x, p) = @. cos(x) - x/p
 u0 = (0, pi/2)
 p = 2
-prob = NonlinearProblem(f, u0, p)
+prob = IntervalNonlinearProblem(f, u0, p)
 solve(prob, Bisection())
 ```
 
@@ -159,7 +159,7 @@ The *insignificant* difference in stopping criteria used by `NonlinearSolve` and
 
 
 ```{julia}
-an = solve(NonlinearProblem{false}(f, u0, p), Bisection())
+an = solve(IntervalNonlinearProblem{false}(f, u0, p), Bisection())
 ar = solve(Roots.ZeroProblem(f, u0), Roots.Bisection(); p=p)
 nextfloat(an[]) == ar, f(an[], p), f(ar, p)
 ```
@@ -418,7 +418,7 @@ prob = OptimizationProblem(sys, u0, p; grad=true, hess=true)
 soln = solve(prob, LBFGS())
 ```
 
-We used an initial guess of $x=4$ above. The `LBFGS` method is a computationally efficient modification of the Broyden-Fletcher-Goldfarb-Shanno algorithm ... It is a quasi-Newton method that updates an approximation to the Hessian using past approximations as well as the gradient."  On this problem it performs similarly to `Newton`, though in general may be preferable for higher-dimensional problems.
+We used an initial guess of $x=4$ above. The `LBFGS` method is a computationally efficient modification of the Broyden-Fletcher-Goldfarb-Shanno algorithm. It is a quasi-Newton method that updates an approximation to the Hessian using past approximations as well as the gradient. On this problem it performs similarly to `Newton`, though in general may be preferable for higher-dimensional problems.
 
 
 ### Two dimensional
@@ -515,7 +515,7 @@ For a simple definite integral, such as $\int_0^\pi \sin(x)dx$, we have:
 
 ```{julia}
 f(x, p) = sin(x)
-prob = IntegralProblem(f, 0.0, pi)
+prob = IntegralProblem(f, 0.0, 1pi)
 soln = solve(prob, QuadGKJL())
 ```
 
@@ -546,7 +546,7 @@ The `Integrals` solution is a bit more verbose, but it is more flexible. For exa
 
 ```{julia}
 f(x, p) = sin.(x)
-prob = IntegralProblem(f, [0.0], [pi])
+prob = IntegralProblem(f, [0.0], [1pi])
 soln = solve(prob, HCubatureJL())
 ```
 
@@ -574,7 +574,7 @@ Consider $d/dp \int_0^\pi \sin(px) dx$. We can do this integral directly to get
 \frac{d}{dp} \int_0^\pi \sin(px)dx
 &= \frac{d}{dp}\left( \frac{-1}{p} \cos(px)\Big\rvert_0^\pi\right)\\
 &= \frac{d}{dp}\left( -\frac{\cos(p\cdot\pi)-1}{p}\right)\\
-&= \frac{\cos(p\cdot \pi) - 1)}{p^2} + \frac{\pi\cdot\sin(p\cdot\pi)}{p}
+&= \frac{\cos(p\cdot \pi) - 1}{p^2} + \frac{\pi\cdot\sin(p\cdot\pi)}{p}
 \end{align*}
 
 Using `Integrals` with `QuadGK` we have:
@@ -583,7 +583,7 @@ Using `Integrals` with `QuadGK` we have:
 ```{julia}
 f(x, p) = sin(p*x)
 function ∫sinpx(p)
-    prob = IntegralProblem(f, 0.0, pi, p)
+    prob = IntegralProblem(f, 0.0, 1pi, p)
     solve(prob, QuadGKJL())
 end
 ```
@@ -614,7 +614,7 @@ The power of a common interface is the ability to swap backends and the uniformi
 #### $f: R^n \rightarrow R$
 
 
-The area under a surface generated by $z=f(x,y)$ over a rectangular region $[a,b]\times[c,d]$ can be readily computed. The two coding implementations require $f$ to be expressed as a function of a vector–*and* a parameter–and the interval to be expressed using two vectors, one for the left endpoints (`[a,c]`) and on for the right endpoints (`[b,d]`).
+The area under a surface generated by $z=f(x,y)$ over a rectangular region $[a,b]\times[c,d]$ can be readily computed. The two coding implementations require $f$ to be expressed as a function of a vector–*and* a parameter–and the interval to be expressed using two vectors, one for the left endpoints (`[a,c]`) and one for the right endpoints (`[b,d]`).
 
 
 For example, the area under the function $f(x,y) = 1 + x^2 + 2y^2$ over $[-1/2, 1/2] \times [-1,1]$ is computed by:
@@ -664,7 +664,7 @@ To compute this transformed integral, we might have:
 ```{julia}
 function vol_sphere(ρ)
     f(rθ, p) = sqrt(ρ^2 - rθ[1]^2) * rθ[1]
-    ls = [0,0]
+    ls = [0.0,0.0]
     rs = [ρ, 2pi]
     prob = IntegralProblem(f, ls, rs)
     solve(prob, HCubatureJL())
@@ -677,7 +677,7 @@ If it is possible to express the region to integrate as $G(R)$ where $R$ is a re
 
 
 $$
-\iint_{G(R)} f(x) dA = \iint_R (f\circ G)(u) |det(J_G(u)| dU
+\iint_{G(R)} f(x) dA = \iint_R (f\circ G)(u) |det(J_G(u))| dU
 $$
 
 turns the integral into the non-rectangular domain $G(R)$ into one over the rectangular domain $R$. The key is to *identify* $G$ and to compute the Jacobian. The latter is simply accomplished with `ForwardDiff.jacobian`.

quarto/alternatives/makie_plotting.qmd

@@ -15,7 +15,7 @@ The [Makie.jl webpage](https://github.com/JuliaPlots/Makie.jl) says
 
 :::{.callout-note}
 ## Examples and tutorials
-`Makie` is a sophisticated plotting package, and capable of an enormous range of plots (cf. [examples](https://makie.juliaplots.org/stable/examples/plotting_functions/).) `Makie` also has numerous [tutorials](https://makie.juliaplots.org/stable/tutorials/) to learn from. These are far more extensive that what is described herein, as this section focuses just on the graphics from calculus.
+`Makie` is a sophisticated plotting package, and capable of an enormous range of plots (cf. [examples](https://makie.juliaplots.org/stable/examples/plotting_functions/).) `Makie` also has numerous [tutorials](https://makie.juliaplots.org/stable/tutorials/) to learn from. These are far more extensive than what is described herein, as this section focuses just on the graphics from calculus.
 
 :::
 
@@ -186,7 +186,7 @@ The curves of calculus are lines. The `lines` command of `Makie` will render a c
 The basic plot of univariate calculus is the graph of a function $f$ over an interval $[a,b]$. This is implemented using a familiar strategy: produce a series of representative values between $a$ and $b$; produce the corresponding $f(x)$ values; plot these as points and connect the points with straight lines.
 
 
-To create regular values between `a` and `b` typically the `range` function or the range operator (`a:h:b`) are employed. The the related `LinRange` function is also an option.
+To create regular values between `a` and `b` typically the `range` function or the range operator (`a:h:b`) are employed. The related `LinRange` function is also an option.
 
 
 For example:

quarto/alternatives/plotly_plotting.qmd

@@ -149,7 +149,7 @@ The difference is solely the specification of the `mode` value, for a line plot
 ### Nothing
 
 
-The line graph plays connect-the-dots with the points specified by paired `x` and `y` values. *Typically*, when and `x` value is `NaN` that "dot" (or point) is skipped. However, `NaN` doesn't pass through the JSON conversion – `nothing` can be used.
+The line graph plays connect-the-dots with the points specified by paired `x` and `y` values. *Typically*, when `x` value is `NaN` that "dot" (or point) is skipped. However, `NaN` doesn't pass through the JSON conversion – `nothing` can be used.
 
 
 ```{julia}
@@ -381,17 +381,17 @@ In the following, to highlight the difference between $f(x) = \cos(x)$ and $p(x)
 xs = range(-1, 1, 100)
 data = [
     Config(
-	    x=xs, y=cos.(xs),
-	    fill = "tonexty",
-	    fillcolor = "rgba(0,0,255,0.25)", # to get transparency
-	    line = Config(color=:blue)
-    ),
-	Config(
 		x=xs, y=[1 - x^2/2 for x ∈ xs ],
 		fill = "tozeroy",
 		fillcolor = "rgba(255,0,0,0.25)", # to get transparency
 		line = Config(color=:red)
-	)
+	  ),
+    Config(
+	    x=xs, y=cos.(xs),
+	    fill = "tonexty",
+	    fillcolor = "rgba(0,0,255,0.25)", # to get transparency
+	    line = Config(color=:blue)
+    )
 ]
 Plot(data)
 ```
@@ -428,7 +428,7 @@ lyt = Config(title = "Main chart title",
 Plot(data, lyt)
 ```
 
-The `xaxis` and `yaxis` keys have many customizations. For example: `nticks` specifies the maximum number of ticks; `range` to set the range of the axis; `type` to specify the axis type from "linear", "log", "date", "category", or "multicategory;" and `visible`
+The `xaxis` and `yaxis` keys have many customizations. For example: `nticks` specifies the maximum number of ticks; `range` to set the range of the axis; `type` to specify the axis type from "linear", "log", "date", "category", or "multicategory"; and `visible`.
 
 
 The aspect ratio of the chart can be set to be equal through the `scaleanchor` key, which specifies another axis to take a value from. For example, here is a parametric plot of a circle: