New ordering of the lecture notes

src/03_Preliminaries.jl

@@ -227,7 +227,7 @@ This indeed can be shown to be the case.
 
 - In comparing with (5) we notice the **monomials** $1 = x^0, x = x^1, x^2, x^3, \ldots, x^n$ to be a possible choice for a basis.
 - Similar to Euclidean vector spaces this is not the only choice of basis and in fact many families of polynomials are known, which are frequently employed as basis functions (e.g. [Lagrange polynomials](https://en.wikipedia.org/wiki/Lagrange_polynomial), [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials), [Hermite polynomials](https://en.wikipedia.org/wiki/Hermite_polynomials), ...)
-- One basis we will discuss in the context of [polynomial interpolation](https://teaching.matmat.org/numerical-analysis/05_Interpolation.html) are Lagrange polynomials, which have the form 
+- One basis we will discuss in the context of [polynomial interpolation](https://teaching.matmat.org/numerical-analysis/07_Interpolation.html) are Lagrange polynomials, which have the form 
   ```math
   \begin{aligned}
   L_{\textcolor{red}{i}}(x) &= \prod_{\stackrel{j=1}{\textcolor{red}{j\neq i}}}^{n+1} \frac{x-x_j}{\textcolor{red}{x_i} - x_j} \\

src/04_Nonlinear_equations.jl

@@ -1393,7 +1393,7 @@ md"""
 	*any iterative procedure*. 
 
 We will consider this aspect further,
-for example in [Iterative methods for linear systems](https://teaching.matmat.org/numerical-analysis/07_Iterative_methods.html).
+for example in [Iterative methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Iterative_methods.html).
 """
 
 # ╔═╡ bdff9554-58b6-466e-9c93-6b1367262b50
@@ -1616,7 +1616,7 @@ end
 md"""
 Note that he linear system $\textbf{A}^{(k)} \textbf{r}^{(k)} = - \textbf{y}^{(k)}$ is solved in Julia using the backslash operator `\`, which employs a numerically more stable algorithm than explicitly computing the inverse `inv(A)` and then applying this to `y`.
 We will discuss these methods in
-[Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html).
+[Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html).
 """
 
 # ╔═╡ 702ffb33-7fbe-4673-aed7-d985a76b455a

src/06_Direct_methods.jl → src/05_Direct_methods.jl

@@ -28,7 +28,7 @@ end
 # ╔═╡ ca2c949f-a6a0-485f-bd52-5dae3b050612
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.pdf)
 """
 
 # ╔═╡ 21c9a859-f976-4a93-bae4-616122712a24
@@ -50,6 +50,9 @@ as well as a right-hand side $\mathbf{b} \in \mathbb{R}^n$.
 As the solution we seek the unknown $\mathbf{x} \in \mathbb{R}^n$.
 """
 
+# ╔═╡ adb09dc3-a074-4b5f-9757-85c05d22ee83
+TODO("polynomial interpolation now comes later")
+
 # ╔═╡ 419d11bf-2561-49ca-a6e7-40c8d8b88b24
 md"""
 - `nmax = ` $(@bind nmax Slider([5, 10, 12, 15]; default=10, show_value=true))
@@ -2328,6 +2331,7 @@ version = "17.4.0+2"
 # ╠═3295f30c-c1f4-11ee-3901-4fb291e0e4cb
 # ╟─21c9a859-f976-4a93-bae4-616122712a24
 # ╟─b3cb31aa-c982-4454-8882-5b840c68df9b
+# ╠═adb09dc3-a074-4b5f-9757-85c05d22ee83
 # ╟─be5d3f98-4c96-4e69-af91-fa2ae5f74af5
 # ╟─419d11bf-2561-49ca-a6e7-40c8d8b88b24
 # ╠═011c25d5-0d60-4729-b200-cdaf3dc89faf

src/07_Iterative_methods.jl → src/06_Iterative_methods.jl

@@ -17,7 +17,7 @@ end
 # ╔═╡ 63bb7fe9-750f-4d2f-9d18-8374b113373e
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/07_Iterative_methods.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/06_Iterative_methods.pdf)
 """
 
 # ╔═╡ 7d9c9392-3aec-4efd-a9ba-d8965687b163
@@ -357,7 +357,7 @@ md"""
 From Theorem 1 we take away that the norm of iteration matrix
 $\|\mathbf{B}\|$ is the crucial quantity to determine not only *if*
 Richardson iterations converge, but also *at which rate*.
-Recall in Lemma 4 of [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html)
+Recall in Lemma 4 of [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html)
 we had the result that for any matrix  $\mathbf{B} \in \mathbb{R}^{m \times n}$
 ```math
 \tag{5}
@@ -469,7 +469,7 @@ obtained by solving the system $\mathbf{A} \mathbf{x}_\ast = \mathbf{b}$ employi
 
 We are thus in exactly the same setting as our
 final section on *Numerical stability* in our discussion
-on [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html)
+on [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html)
 where instead of solving $\mathbf{A} \mathbf{x}_\ast = \mathbf{b}$
 we are only able to solve the perturbed system
 $\mathbf{A} \widetilde{\textbf{x}} = \widetilde{\mathbf{b}}$.
@@ -478,7 +478,7 @@ $\mathbf{A} \widetilde{\textbf{x}} = \widetilde{\mathbf{b}}$.
 # ╔═╡ 55a69e52-002f-40dc-8830-7fa16b7af081
 md"""
 We can thus directly apply Theorem 2
-from [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html), which states that
+from [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html), which states that
 ```math
 \frac{\|\mathbf{x}_\ast - \widetilde{\mathbf{x}} \|}{\| \mathbf{x}_\ast \|}
 ≤ κ(\mathbf{A}) 
@@ -839,7 +839,7 @@ Importantly there is thus a **relation between optimisation problems** and **sol
 
 # ╔═╡ bf9a171a-8aa4-4f21-bde3-56ccef40de24
 md"""
-SPD matrices are not unusual. For example, recall that in polynomial regression problems (see least-squares problems in [Interpolation](https://teaching.matmat.org/numerical-analysis/05_Interpolation.html)),
+SPD matrices are not unusual. For example, recall that in polynomial regression problems (see least-squares problems in [Interpolation](https://teaching.matmat.org/numerical-analysis/07_Interpolation.html)),
 where we wanted to find the best polynomial through the points
 $(x_i, y_i)$ for $i=1, \ldots n$ by minimising the least-squares error,
 we had to solve the *normal equations*

src/05_Interpolation.jl → src/07_Interpolation.jl

@@ -30,7 +30,7 @@ end
 # ╔═╡ 46b46b8e-b388-44e1-b2d8-8d7cfdc3b475
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/05_Interpolation.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/07_Interpolation.pdf)
 """
 
 # ╔═╡ 61e5ef66-a213-4b23-9406-9cc63a58104c
@@ -692,6 +692,9 @@ This is an example of **exponential convergence**: The error of the approximatio
 The **graphical characterisation** is similar to the iterative schemes we discussed in the previous chapter: We employ a **semilog plot** (using a linear scale for $n$ and a logarithmic scale for the error), where exponential convergence is characterised by a straight line:
 """
 
+# ╔═╡ 21c98bd4-b3eb-4406-bcd2-0abfbeb9bb93
+TODO("'previous chapter' remark likely outdated after pushing interpolation back")
+
 # ╔═╡ d4cf71ef-576d-4900-9608-475dbd4d933a
 let
 	fine = range(-1.0, 1.0; length=3000)
@@ -730,6 +733,9 @@ is one of the **desired properties**.
 	  * If the error scales as $α C^{n}$ where $n$ is some accuracy parameter (with larger $n$ giving more accurate results), then we say the scheme has **exponential convergence**.
 """
 
+# ╔═╡ 647f96ee-c0ad-4bd8-9de1-f24a7dcf6b24
+TODO("'Last chapter' reference is likely outdated after pushing interpolation back")
+
 # ╔═╡ a15750a3-3507-4ee1-8b9a-b7d6a3dcea46
 md"""
 ### Stability of polynomial interpolation
@@ -827,7 +833,7 @@ md"""
 
 	Since for Chebyshev nodes $\Lambda_n$ stays relatitvely small, we would call Chebyshev interpolation **well-conditioned**. In contrast interpolation using equally spaced nodes is **ill-conditioned** as the condition number $\Lambda_n$ can get very large, thus **even small input errors can amplify** and **drastically reduce the accuracy** of the obtained polynomial.
 
-	We will meet other condition numbers later in the lecture, e.g. in [Iterative methods for linear systems](https://teaching.matmat.org/numerical-analysis/07_Iterative_methods.html).
+	We will meet other condition numbers later in the lecture, e.g. in [Iterative methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Iterative_methods.html).
 """
 
 # ╔═╡ 5e19f1a7-985e-4fb7-87c4-5113b5615521
@@ -1954,7 +1960,7 @@ md"""
   * The typical approach are **Chebyshev nodes**
   * These lead to **exponential convergence**
 
-Notice that all of these problems lead to linear systems $\textbf A \textbf x = \textbf b$ that we need to solve. How this can me done numerically we will see in [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html).
+Notice that all of these problems lead to linear systems $\textbf A \textbf x = \textbf b$ that we need to solve. How this can me done numerically we will see in [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html).
 """
 
 # ╔═╡ 2240f8bc-5c0b-450a-b56f-2b53ca66bb03
@@ -3307,8 +3313,10 @@ version = "1.4.1+2"
 # ╟─25b82572-b27d-4f0b-9be9-323cd4e3ce7a
 # ╟─c38b9e48-98bb-4b9c-acc4-7375bbd39ade
 # ╟─479a234e-1ce6-456d-903a-048bbb3de65a
+# ╠═21c98bd4-b3eb-4406-bcd2-0abfbeb9bb93
 # ╟─d4cf71ef-576d-4900-9608-475dbd4d933a
 # ╟─56685887-7866-446c-acdb-2c20bd11d4cd
+# ╠═647f96ee-c0ad-4bd8-9de1-f24a7dcf6b24
 # ╟─a15750a3-3507-4ee1-8b9a-b7d6a3dcea46
 # ╟─7f855423-72ac-4e6f-92bc-73c12e5007eb
 # ╟─eaaf2227-1a19-4fbc-a5b4-45503e832280

src/09_Numerical_integration.jl → src/08_Numerical_integration.jl

@@ -18,7 +18,7 @@ end
 # ╔═╡ d34833b7-f375-40f7-a7a6-ab925d736320
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/09_Numerical_integration.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/08_Numerical_integration.pdf)
 """
 
 # ╔═╡ 47114a9b-0e74-4e48-bb39-b49f526f1e9b
@@ -107,7 +107,7 @@ and **then integrate that** instead of $f$ itself.
 Since the integration of the polynomial is essentially exact,
 the error of such a scheme is **dominated by the error of the polynomial
 interpolation**.
-Recall the [chapter on Interpolation](https://teaching.matmat.org/numerical-analysis/05_Interpolation.html), where we noted polynomials
+Recall the [chapter on Interpolation](https://teaching.matmat.org/numerical-analysis/07_Interpolation.html), where we noted polynomials
 through equispaced nodes to become numerically unstable and
 possibly inaccurate for large $n$ due to Runge's phaenomenon.
 
@@ -211,7 +211,7 @@ end
 
 # ╔═╡ a1d83cb2-6e0d-4a53-a11f-60dc020249d4
 md"""
-Recall that in Theorem 4 of [chapter 05 (Interpolation)](https://teaching.matmat.org/numerical-analysis/05_Interpolation.html) we found that
+Recall that in Theorem 4 of [chapter 07 (Interpolation)](https://teaching.matmat.org/numerical-analysis/07_Interpolation.html) we found that
 the piecewise polynomial interpolation shows quadratic convergence
 ```math
 \|f - p_{1,h}\|_\infty \leq α h^2 \| f'' \|_\infty,

src/10_Numerical_differentiation.jl → src/09_Numerical_differentiation.jl

@@ -31,7 +31,7 @@ end
 # ╔═╡ 4103c9d2-ef89-4c65-be3f-3dab59d1cc47
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/10_Numerical_differentiation.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/09_Numerical_differentiation.pdf)
 """
 
 # ╔═╡ e9151d3f-8d28-4e9b-add8-43c713f6f068

src/12_Boundary_value_problems.jl → src/10_Boundary_value_problems.jl

@@ -29,7 +29,7 @@ end
 # ╔═╡ b72b45ad-6191-40cb-9e9f-950bf1bfe212
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/12_Boundary_value_problems.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/10_Boundary_value_problems.pdf)
 """
 
 # ╔═╡ 206ae56c-fcfa-4d6f-93e4-30f03dee8f90
@@ -176,7 +176,7 @@ u(0) &= b_0, \quad u(L) = b_L,
 \right.
 ```
 were $b_0, b_L \in \mathbb{R}$.
-Similar to our approach when [solving initial value problems (chapter 11)](https://teaching.matmat.org/numerical-analysis/11_Initial_value_problems.html)
+Similar to our approach when [solving initial value problems (chapter 12)](https://teaching.matmat.org/numerical-analysis/12_Initial_value_problems.html)
 we **divide the full interval $[0, L]$ into $N+1$ subintervals** $[x_j, x_{j+1}]$
 of uniform size $h$, i.e. 
 ```math
@@ -185,6 +185,9 @@ x_j = j\, h \quad j = 0, \ldots, N+1, \qquad h = \frac{L}{N+1}.
 Our goal is thus to find approximate points $u_j$ such that $u_j ≈ u(x_j)$ at the nodes $x_j$.
 """
 
+# ╔═╡ 782dff7d-76f5-4977-98cb-81881a05331a
+TODO("IVP is now after BCP, adjust reference accordingly")
+
 # ╔═╡ 82788dfd-3462-4f8e-b0c8-9e196dac23a9
 md"""
 Due to the Dirichlet boundary conditions $u(0) = b_0$ and $u(L) = b_L$.
@@ -201,7 +204,7 @@ These internal nodes $u(x_j)$ need to satisfy
 - \frac{\partial^2 u}{\partial x^2}(x_j) = f(x_j) \qquad \forall\, 1 ≤ j ≤ N.
 ```
 As the derivatives of $u$ are unknown to us we employ a
-**[central finite-difference formula](https://teaching.matmat.org/numerical-analysis/10_Numerical_differentiation.html)**
+**[central finite-difference formula](https://teaching.matmat.org/numerical-analysis/09_Numerical_differentiation.html)**
 to replace this derivative by the approximation
 ```math
 \tag{3}
@@ -288,7 +291,7 @@ which is to be solved for the unknows $\mathbf{u}$.
 # ╔═╡ c21502ce-777f-491a-a536-ff499fc172fc
 md"""
 We notice that $\mathbf{A}$ is **symmetric and tridiagonal**. Additionally one can show $\mathbf{A}$ to be **positive definite**.
-Problem (8) can therefore be **efficiently solved** using [direct methods based on (sparse) LU factorisation (chapter 6)](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html) or an [iterative approaches (chapter 7)](https://teaching.matmat.org/numerical-analysis/07_Iterative_methods.html), e.g. the conjugate gradient method.
+Problem (8) can therefore be **efficiently solved** using [direct methods based on (sparse) LU factorisation (chapter 5)](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html) or an [iterative approaches (chapter 6)](https://teaching.matmat.org/numerical-analysis/06_Iterative_methods.html), e.g. the conjugate gradient method.
 """
 
 # ╔═╡ c2bb42b3-4fee-4ad4-84c0-06f58c7f7665
@@ -505,7 +508,7 @@ md"""
 While initially the convergence thus nicely follows the expected convergence curve, **for larger $N$ the convergence degrades and the error starts increasing again**.
 
 Similar to our discussion on numerical stability
-in the [chapter on numerical differentiation](https://teaching.matmat.org/numerical-analysis/10_Numerical_differentiation.html)
+in the [chapter on numerical differentiation](https://teaching.matmat.org/numerical-analysis/09_Numerical_differentiation.html)
 this error plot is the result of a balance between two error contributions:
 - The **discretisation error** due to the choice of $N$, where as $N$ gets larger
   this error **decreases** as $O(N^{-2})$.
@@ -1122,7 +1125,7 @@ md"""
 A widely employed set of basis functions for Galerkin approximations
 are the hat functions $φ_i = H_i$,
 which we already discussed in the chapter
-on [Interpolation (chapter 5)](https://teaching.matmat.org/numerical-analysis/05_Interpolation.html).
+on [Interpolation (chapter 7)](https://teaching.matmat.org/numerical-analysis/07_Interpolation.html).
 Recall, that given a set of nodes $x_0 < x_1 < \cdots < x_{n}$
 the hat functions are defined as
 ```math
@@ -2656,6 +2659,7 @@ version = "1.4.1+2"
 # ╟─3e10cf8e-d5aa-4b3e-a7be-12ccdc2f3cf7
 # ╟─7fd851e6-3180-4008-a4c0-0e08edae9954
 # ╟─52c7ce42-152d-40fd-a910-78f755fcae47
+# ╠═782dff7d-76f5-4977-98cb-81881a05331a
 # ╟─82788dfd-3462-4f8e-b0c8-9e196dac23a9
 # ╟─d43ecff3-89a3-4edd-95c2-7262e317ce29
 # ╟─1fb53091-89c8-4f70-ab4b-ca2371b830b2

src/08_Eigenvalue_problems.jl → src/11_Eigenvalue_problems.jl

@@ -30,7 +30,7 @@ end
 # ╔═╡ 34beda8f-7e5f-42eb-b32c-73cfc724062e
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/08_Eigenvalue_problems.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/11_Eigenvalue_problems.pdf)
 """
 
 # ╔═╡ 13298dc4-9800-476d-9474-182359a7671b
@@ -59,7 +59,7 @@ then we find
 \sqrt{λ_\text{max}(\mathbf A^T \mathbf A)} \, \| \mathbf x \|
 = λ_\text{max}(\mathbf A) \, \| \mathbf x \|
 ```
-where we used the inequalities introduced at the end of  [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html).
+where we used the inequalities introduced at the end of  [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html).
 We note that the **largest eigenvalue of $\mathbf A$**
 provides a **bound to the action of $\mathbf{A}$**.
 """
@@ -849,7 +849,7 @@ the iterative loop.
 Since for dense matrices
 computing the factorisation scales $O(n^3)$,
 but solving linear systems based on the factorisation only scales $O(n^2)$
-([recall chapter 6](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html)), this reduces the cost per iteration.
+([recall chapter 5](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html)), this reduces the cost per iteration.
 """
 
 # ╔═╡ 8e01a98d-c49f-43b3-9681-07d8e4b7f12a

src/11_Initial_value_problems.jl → src/12_Initial_value_problems.jl

@@ -31,7 +31,7 @@ end
 # ╔═╡ ba9b6172-0234-442c-baaa-876b12f689bd
 md"""
 !!! info ""
-    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/11_Initial_value_problems.pdf)
+    [Click here to view the PDF version.](https://teaching.matmat.org/numerical-analysis/12_Initial_value_problems.pdf)
 """
 
 # ╔═╡ d8406b01-e36f-4953-a5af-cd563005c2a1
@@ -293,7 +293,7 @@ Our task is to find $u(t_{n+1})$.
 md"""
 We make progress by approximating the dervative of $u$
 using one of the finite differences formulas
-discussed in the chapter on [Numerical differentiation](https://teaching.matmat.org/numerical-analysis/10_Numerical_differentiation.html).
+discussed in the chapter on [Numerical differentiation](https://teaching.matmat.org/numerical-analysis/09_Numerical_differentiation.html).
 
 The simplest approach is to employ forward finite differences, i.e.
 ```math