diff --git a/src/03_Preliminaries.jl b/src/03_Preliminaries.jl index e633307..3bfdc7f 100644 --- a/src/03_Preliminaries.jl +++ b/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} \\ diff --git a/src/04_Nonlinear_equations.jl b/src/04_Nonlinear_equations.jl index 18fe8cd..c7e8ac7 100644 --- a/src/04_Nonlinear_equations.jl +++ b/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 diff --git a/src/06_Direct_methods.jl b/src/05_Direct_methods.jl similarity index 99% rename from src/06_Direct_methods.jl rename to src/05_Direct_methods.jl index 49190e0..dac776f 100644 --- a/src/06_Direct_methods.jl +++ b/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 diff --git a/src/07_Iterative_methods.jl b/src/06_Iterative_methods.jl similarity index 99% rename from src/07_Iterative_methods.jl rename to src/06_Iterative_methods.jl index 321dd1d..523de39 100644 --- a/src/07_Iterative_methods.jl +++ b/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* diff --git a/src/05_Interpolation.jl b/src/07_Interpolation.jl similarity index 99% rename from src/05_Interpolation.jl rename to src/07_Interpolation.jl index d710229..cc9cc0d 100644 --- a/src/05_Interpolation.jl +++ b/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 diff --git a/src/09_Numerical_integration.jl b/src/08_Numerical_integration.jl similarity index 99% rename from src/09_Numerical_integration.jl rename to src/08_Numerical_integration.jl index 5822e3c..4cc4140 100644 --- a/src/09_Numerical_integration.jl +++ b/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, diff --git a/src/10_Numerical_differentiation.jl b/src/09_Numerical_differentiation.jl similarity index 99% rename from src/10_Numerical_differentiation.jl rename to src/09_Numerical_differentiation.jl index 165300f..6eb5bb8 100644 --- a/src/10_Numerical_differentiation.jl +++ b/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 diff --git a/src/12_Boundary_value_problems.jl b/src/10_Boundary_value_problems.jl similarity index 99% rename from src/12_Boundary_value_problems.jl rename to src/10_Boundary_value_problems.jl index 2e78338..2cc7536 100644 --- a/src/12_Boundary_value_problems.jl +++ b/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 diff --git a/src/08_Eigenvalue_problems.jl b/src/11_Eigenvalue_problems.jl similarity index 99% rename from src/08_Eigenvalue_problems.jl rename to src/11_Eigenvalue_problems.jl index 783dea2..93993f2 100644 --- a/src/08_Eigenvalue_problems.jl +++ b/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 diff --git a/src/11_Initial_value_problems.jl b/src/12_Initial_value_problems.jl similarity index 99% rename from src/11_Initial_value_problems.jl rename to src/12_Initial_value_problems.jl index 99a1448..ee124b9 100644 --- a/src/11_Initial_value_problems.jl +++ b/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 diff --git a/src/sidebar.md b/src/sidebar.md index 5ceda9e..3ccbe97 100644 --- a/src/sidebar.md +++ b/src/sidebar.md @@ -4,11 +4,11 @@ 1. [The Julia programming language](https://teaching.matmat.org/numerical-analysis/02_Julia.html) 1. [Revision and preliminaries](https://teaching.matmat.org/numerical-analysis/03_Preliminaries.html) 1. [Root finding and fixed-point problems](https://teaching.matmat.org/numerical-analysis/04_Nonlinear_equations.html) -1. [Interpolation](https://teaching.matmat.org/numerical-analysis/05_Interpolation.html) -1. [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Direct_methods.html) -1. [Iterative methods for linear systems](https://teaching.matmat.org/numerical-analysis/07_Iterative_methods.html) -1. [Eigenvalue problems](https://teaching.matmat.org/numerical-analysis/08_Eigenvalue_problems.html) -1. [Numerical integration](https://teaching.matmat.org/numerical-analysis/09_Numerical_integration.html) -1. [Numerical differentiation](https://teaching.matmat.org/numerical-analysis/10_Numerical_differentiation.html) -1. [Initial value problems](https://teaching.matmat.org/numerical-analysis/11_Initial_value_problems.html) -1. [Boundary value problems](https://teaching.matmat.org/numerical-analysis/12_Boundary_value_problems.html) +1. [Direct methods for linear systems](https://teaching.matmat.org/numerical-analysis/05_Direct_methods.html) +1. [Iterative methods for linear systems](https://teaching.matmat.org/numerical-analysis/06_Iterative_methods.html) +1. [Interpolation](https://teaching.matmat.org/numerical-analysis/07_Interpolation.html) +1. [Numerical integration](https://teaching.matmat.org/numerical-analysis/08_Numerical_integration.html) +1. [Numerical differentiation](https://teaching.matmat.org/numerical-analysis/09_Numerical_differentiation.html) +1. [Boundary value problems](https://teaching.matmat.org/numerical-analysis/10_Boundary_value_problems.html) +1. [Eigenvalue problems](https://teaching.matmat.org/numerical-analysis/11_Eigenvalue_problems.html) +1. [Initial value problems](https://teaching.matmat.org/numerical-analysis/12_Initial_value_problems.html)