lectures/lecture-15/lecture-15.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Lecture 15:  Great Iterative Methods"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Previous lecture\n",
    "\n",
    "- Concept of **iterative methods** for linear systems:\n",
    "    - Richardson iteration and its convergence \n",
    "    - Chebyshev iteration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Plan for this part of class\n",
    "\n",
    "- Lanczos and Arnoldi orthogonalization of Krylov subspaces, optimality result for Krylov subspaces\n",
    "- Main iterative methods: conjugate gradient, GMRES, etc\n",
    "- Convergence estimates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Solution of linear systems and minimization of functionals\n",
    "\n",
    "- Instead of solving a linear system, we can minimize the **residual:**\n",
    "\n",
    "$$R(x) = \\Vert A x - f \\Vert^2_2.$$\n",
    "\n",
    "- The condition $\\nabla R(x) = 0$ gives\n",
    "\n",
    "$$A^* A x = A^* f,$$\n",
    "\n",
    "thus it has squared condition number, so direct minimization of the residual by standard optimization methods is rarely used.\n",
    "\n",
    "- For the symmetric positive definite case there is a much simpler functional."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Energy functional\n",
    "\n",
    "Let $A = A^* > 0$, then the following functional \n",
    "\n",
    "$$\\Phi(x) = (Ax, x)  - 2(f, x)$$\n",
    "\n",
    "is called **energy functional**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Properties of energy functional\n",
    "- It is strictly convex (check!)\n",
    "\n",
    "$$ \\Phi(\\alpha x + (1 - \\alpha)y) < \\alpha \\Phi(x) + (1 - \\alpha) \\Phi(y)$$\n",
    "\n",
    "- Since it is strictly convex, it has unique local minimum, which is also global\n",
    "\n",
    "- Its global minimum $x_*$ satisfies\n",
    "\n",
    "$$A x_* = f.$$\n",
    "\n",
    "Indeed, \n",
    "\n",
    "$$\\nabla \\Phi = 2(Ax - f).$$ \n",
    "\n",
    "and the first order optimality condition $\\nabla \\Phi (x_*) = 0$ yields\n",
    "\n",
    "$$A x_* = f.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Approximation of the solution by a subspace\n",
    "\n",
    "- Given a linear $M$-dimensional subspace $\\{y_1, \\dots, y_M\\}$, we want to find an approximate solution in this basis, i.e. \n",
    "\n",
    "$$A x \\approx f, \\quad x = x_0 +  \\sum_{k=1}^M c_k y_k,$$\n",
    "\n",
    "where $c$ is the vector of coefficients.\n",
    "\n",
    "- In the symmetric positive definite case  we need to minimize \n",
    "\n",
    "$$(Ax, x) - 2(f, x)$$ \n",
    "\n",
    "subject to $$x = x_0 + Y c,$$\n",
    "\n",
    "where $Y=[y_1,\\dots,y_M]$ is $n \\times M$ and vector $c$ has length $M$.\n",
    "\n",
    "- Using the representation of $x$, we have the following minimization for $c$:\n",
    "\n",
    "$$\\widehat{\\Phi}(c) = (A Y c, Y c) + 2 (Y^*Ax_0, c) - 2(f, Y c) = (Y^* A Y c, c) - 2(Y^* (f - Ax_0), c).$$\n",
    "\n",
    "- Note that this is the same functional, but for the **Galerkin projection** of $A$\n",
    "\n",
    "$$Y^* A Y c = Y^* (f - Ax_0) = Y^* r_0,$$\n",
    "\n",
    "which is an $M \\times M$  linear system with symmetric positive definite matrix if $Y$ has full column rank.\n",
    "\n",
    "But how to choose $Y$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Selection of the subspace: random projection\n",
    "\n",
    "- We can generate $Y$ with random numbers and then orthogonalize\n",
    "- What quality of $x$ will we get in this case?\n",
    "- How the derived quality relates to the conditioning of the matrix?\n",
    "- Selection of the proper random matrix is important topic in dimensionality reduction theory and relates to [random projection](http://cs-people.bu.edu/evimaria/cs565/kdd-rp.pdf) approach "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(30, 30)\n",
      "0.0004114360569253815\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "n = 100\n",
    "A = np.random.randn(n, n)\n",
    "Q, _ = np.linalg.qr(A)\n",
    "k = 70\n",
    "A = Q.T @ np.diag([1e-6] * k + list(np.random.rand(n-k))) @ Q\n",
    "x_true = np.random.randn(n)\n",
    "rhs = A @ x_true\n",
    "M = n - k\n",
    "Y = np.random.randn(n, M)\n",
    "A_proj = Y.T @ A @ Y\n",
    "rhs_proj = Y.T @ rhs\n",
    "print(A_proj.shape)\n",
    "c = np.linalg.solve(A_proj, rhs_proj)\n",
    "x_proj = Y @ c\n",
    "print(np.linalg.norm(A @ x_proj - rhs) / np.linalg.norm(rhs))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Selection of the subspace: Krylov subspace\n",
    "\n",
    "In the Krylov subspace we generate the whole subspace from a single vector $r_0 = f - Ax_0$:\n",
    "\n",
    "$$y_0\\equiv k_0 = r_0, \\quad y_1\\equiv k_1 = A r_0, \\quad y_2\\equiv k_2 = A^2 r_0, \\ldots, \\quad y_{M-1}\\equiv k_{M-1} = A^{M-1} r_0.$$\n",
    "\n",
    "This gives the **Krylov subpace** of the $M$-th order\n",
    "\n",
    "$$\\mathcal{K}_M(A, r_0) = \\mathrm{Span}(r_0, Ar_0, \\ldots, A^{M-1} r_0).$$\n",
    "\n",
    "- It is known to be quasi-optimal space given only matrix-vector product operation.\n",
    "- Key reference here is \"On the numerical solution of equation by which are determined in technical problems the frequencies of small vibrations of material systems\",  A. N. Krylov, 1931, [text in russian](http://www.mathnet.ru/links/a2431bc65c0764da9a1ae95c73f741f6/im5215.pdf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Solution $x_*$ lies in the Krylov subspace: $x_* \\in \\mathcal{K}_n(A, f)$\n",
    "\n",
    "- According to [Cayley–Hamilton theorem](https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem): $p(A) = 0$, where $p(\\lambda) = \\det(A - \\lambda I)$\n",
    "- $p(A)f = A^nf + a_1A^{n-1}f + \\ldots + a_{n-1}Af + a_n f = 0$\n",
    "- $A^{-1}p(A)f = A^{n-1}f + a_1A^{n-2}f + \\ldots + a_{n-1}f + a_nA^{-1}f = 0$\n",
    "- $x_* = A^{-1}f = -\\frac{1}{a_n}(A^{n-1}f + a_1A^{n-2}f + \\ldots + a_{n-1}f)$\n",
    "- Thus, $x_* \\in \\mathcal{K}_n(A, f)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ill-conditioned of the natural basis\n",
    "\n",
    "The natural basis in the Krylov subspace is very **ill-conditioned**, since \n",
    "\n",
    "$$k_i = A^i r_0 \\rightarrow \\lambda_\\max^i v,$$\n",
    "\n",
    "where $v$ is the eigenvector, corresponding to the maximal eigenvalue of $A$, i.e. $k_i$ become more and more collinear for large $i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Condition number = 610650129.027377\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.sparse as spsp\n",
    "%matplotlib inline\n",
    "\n",
    "n = 100\n",
    "ex = np.ones(n);\n",
    "A = spsp.spdiags(np.vstack((-ex,  2*ex, -ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "f = np.ones(n)\n",
    "x0 = np.random.randn(n)\n",
    "\n",
    "subspace_order = 10\n",
    "krylov_vectors = np.zeros((n, subspace_order))\n",
    "krylov_vectors[:, 0] = f - A.dot(x0)\n",
    "for i in range(1, subspace_order):\n",
    "    krylov_vectors[:, i] = A.dot(krylov_vectors[:, i-1])\n",
    "    \n",
    "s = np.linalg.svd(krylov_vectors, compute_uv=False)\n",
    "print(\"Condition number = {}\".format(s.max() / s.min()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Solution:** Compute orthogonal basis in the Krylov subspace."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Good basis in a Krylov subspace\n",
    "\n",
    "In order to have stability, we first orthogonalize the vectors from the Krylov subspace using **Gram-Schmidt** orthogonalization process (or, QR-factorization).\n",
    "\n",
    "$$K_j = \\begin{bmatrix} r_0 & Ar_0 & A^2 r_0 & \\ldots & A^{j-1} r_0\\end{bmatrix} = Q_j R_j, $$\n",
    "\n",
    "and the solution will be approximated as \n",
    "\n",
    "$$x \\approx x_0 + Q_j c.$$ \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Short way to Arnoldi relation\n",
    "\n",
    "**Statement.** The Krylov matrix $K_j$ satisfies an important recurrent relation (called **Arnoldi relation**)\n",
    "\n",
    "$$A Q_j = Q_j H_j + h_{j, j-1} q_j e^{\\top}_{j-1},$$\n",
    "\n",
    "where $H_j$ is upper Hessenberg, and  $Q_{j+1} = [q_0,\\dots,q_j]$ has orthogonal columns that spans columns of $K_{j+1}$.\n",
    "\n",
    "Let us prove it (consider $j = 3$ for simplicity):\n",
    "\n",
    "$$A \\begin{bmatrix} k_0 & k_1 & k_2 \\end{bmatrix} = \\begin{bmatrix} k_1 & k_2 & k_3 \\end{bmatrix} = \\begin{bmatrix} k_0 & k_1 & k_2 \\end{bmatrix} \\begin{bmatrix} 0 & 0 & \\alpha_0 \\\\ 1 & 0  & \\alpha_1 \\\\ 0 & 1  & \\alpha_2 \\\\ \\end{bmatrix} + \\begin{bmatrix} 0 & 0 & k_3  - \\alpha_0 k_0 - \\alpha_1 k_1 - \\alpha_2 k_2 \\end{bmatrix}, $$\n",
    " \n",
    "where $\\alpha_s$ will be selected later. \n",
    "Denote $\\widehat{k}_3 = k_3  - \\alpha_0 k_0 - \\alpha_1 k_1 - \\alpha_2 k_2$.\n",
    " \n",
    " In the matrix form,\n",
    " \n",
    " $$A K_3 = K_3 Z + \\widehat k_3 e^{\\top}_2,$$\n",
    " \n",
    " where $Z$ is the **lower shift** matrix with the last column $(\\alpha_0,\\alpha_1,\\alpha_2)^T$, and $e_2$ is the last column of the identity matrix.\n",
    " \n",
    " Let \n",
    " \n",
    " $$K_3 = Q_3 R_3$$ \n",
    " \n",
    " be the QR-factorization. Then,\n",
    " \n",
    " $$A Q_3 R_3 = Q_3 R_3 Z + \\widehat{k}_3 e^{\\top}_2,$$\n",
    " \n",
    " $$ A Q_3 = Q_3 R_3 Z R_3^{-1} + \\widehat{k}_3 e^{\\top}_2 R_3^{-1}.$$\n",
    "\n",
    "\n",
    "Note that \n",
    "\n",
    "$$e^{\\top}_2 R_3^{-1} = \\begin{bmatrix} 0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix} * & * & * \\\\ 0 & * & * \\\\ 0 & 0 & * \\end{bmatrix}  = \\gamma e^{\\top}_2,$$\n",
    " and\n",
    " \n",
    " $$R_3 Z R_3^{-1} = \\begin{bmatrix} * & * & * \\\\* & * & * \\\\  0 & * & * \\\\ \\end{bmatrix},$$      \n",
    "\n",
    "in the general case it will be an **upper Hessenberg matrix** $H$, i.e. a matrix \n",
    "that \n",
    "\n",
    "$$H_{ij} = 0, \\quad \\mbox{if } i > j + 1.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## (Almost) Arnoldi relation\n",
    "\n",
    "Let $Q_j$ be the orthogonal basis in the Krylov subspace, then we have almost the Arnoldi relation\n",
    "\n",
    "$$A Q_j = Q_j H_j +  \\gamma\\widehat{k}_j e^{\\top}_{j-1},$$\n",
    "\n",
    "where $H_j$ is an upper Hessenberg matrix, and \n",
    "\n",
    "$$\\widehat{k}_j = k_j - \\sum_{s=0}^{j-1} \\alpha_s k_s.$$\n",
    "\n",
    "We select $\\alpha_s$ in such a way that \n",
    "\n",
    "$$Q^*_j \\widehat{k}_j = 0.$$\n",
    "\n",
    "Then, $\\widehat{k}_j = h_{j, j-1} q_j,$ where $q_j$ is the last column of $Q_{j+1}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Arnoldi relation: final formula\n",
    "\n",
    "We have \n",
    "\n",
    "$$A Q_j = Q_j H_j + h_{j, j-1} q_j e^{\\top}_{j-1}.$$\n",
    "\n",
    "- This is the crucial formula for the efficient generation of such subspaces.\n",
    "\n",
    "- For non-symmetric case, it is just modified Gram-Schmidt.\n",
    "\n",
    "- For the symmetric case, we have a much simpler form (Lanczos process)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Lanczos process\n",
    "\n",
    "If $A = A^*$, then\n",
    "\n",
    "$$Q^*_j A Q_j = H_j, $$\n",
    "\n",
    "thus $H_j$ is hermitian, and thus it is **tridiagonal**, $H_j = T_j$.\n",
    "\n",
    "This gives a short-term recurrence relation to generate the Arnoldi vectors $q_j$ without **full** orthogonalization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Lanczos process (2)\n",
    "\n",
    "$$ A Q_j = Q_j T_j + t_{j, j-1} q_j e^{\\top}_{j-1}.$$\n",
    "\n",
    "\n",
    "In order to get $q_j$, we need to compute just the last column of \n",
    "\n",
    "$$t_{j, j-1} q_j = (A Q_j - Q_j T_j) e_{j-1} = A q_{j-1} - t_{j-1, j-1} q_{j-1} - t_{j-2, j-1} q_{j-2}. $$\n",
    "\n",
    "The coefficients $\\alpha_j = t_{j-1, j-1}$ and $\\beta_j = t_{j-2, j-1}$ can be recovered from orthogonality constraints \n",
    "\n",
    "$(q_j, q_{j-1}) = 0, \\quad (q_j, q_{j-2}) = 0$\n",
    "\n",
    "**All the other constraints will be satisfied automatically!!**\n",
    "\n",
    "And we only need to store two vectors to get the new one."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## From direct Lanczos method to the conjugate gradient\n",
    "\n",
    "We can now get from the Lanczos recurrence to the famous **conjugate gradient** method.\n",
    "\n",
    "We have for $A = A^* > 0$\n",
    "\n",
    "$$A Q_j = Q_j T_j + T_{j, j-1} q_j.$$\n",
    "\n",
    "Recall that when we minimize energy functional in basis $Y$ we get a system $Y^* A Y c = Y^* f,$.\n",
    "Here $Y = Q_j$, so the approximate solution of $Ax \\approx f$ with $x_j = x_0 + Q_j c_j$ can be found by solving a small system\n",
    "\n",
    "$$Q^*_j A Q_j c_j = T_j c_j = Q^*_j r_0 .$$\n",
    "\n",
    "Since $f$ is the first Krylov subspace, then \n",
    "**Note!!!** (recall what the first column in $Q_j$ is)\n",
    "\n",
    "$$Q^*_j r_0  = \\Vert r_0 \\Vert_2 e_0 = \\gamma e_0.$$\n",
    "\n",
    "We have a tridiagonal system of equations for $c$:\n",
    "\n",
    "$$T_j c_j = \\gamma e_0$$\n",
    "\n",
    "and $x_j = Q_j c_j$.\n",
    "\n",
    "We could stop at this point, but we want short recurrent formulas instead of solving linear system with matrix $T_j$ at each step."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Derivation of the following update formulas is not required on the oral exam!**\n",
    "\n",
    "- Since $A$ is positive definite, $T_j$ is also positive definite, and it allows an LU decomposition\n",
    "\n",
    "- $T_j = L_j U_j$, where $L_j$ is a bidiagonal matrix with ones on the diagonal, $U_j$ is a upper bidiagonal matrix.\n",
    "\n",
    "$$ T_j = \\begin{bmatrix} a_1 & b_1 &  & \\\\ b_1 & a_2 & b_2 & \\\\ & \\ddots & \\ddots & \\ddots & \\\\ & & b_{j-1} & a_{j-1} & b_j \\\\ & & & b_j & a_j \\end{bmatrix} = \\begin{bmatrix} 1 & &  & \\\\ c_1 & 1 &  & \\\\ & \\ddots & \\ddots &  & \\\\ & & c_{j-1} & 1 & \\\\ & & & c_j & 1 \\end{bmatrix} \\begin{bmatrix} d_1 & b_1 &  & \\\\ & d_1 & b_2 & \\\\ & & \\ddots & \\ddots & \\\\ & & & d_{j-1} & b_j \\\\ & & & & d_j \\end{bmatrix} $$ \n",
    "\n",
    "- We need to define one subdiagonal in $L$ (with elements $c_1, \\ldots, c_{j-1}$), main diagonal of $U_j$ (with elements $d_0, \\ldots, d_{j-1}$ and superdiagonal of $U_j$ (with elements $b_1, \\ldots, b_{j-1}$).\n",
    "\n",
    "- They have convenient recurrences:\n",
    "\n",
    "$$c_i = b_i/d_{i-1}, \\quad d_i = \\begin{cases} a_1, & \\mbox{if } i = 1, \\\\\n",
    "a_i - c_i b_i, & \\mbox{if } i > 1. \\end{cases}$$\n",
    "\n",
    "- For the solution we have\n",
    "\n",
    "$$x_j = Q_j T^{-1}_j \\gamma e_0  = \\gamma Q_j (L_j U_j)^{-1} e_0  = \\gamma Q_j U^{-1}_j L^{-1}_j e_0.$$\n",
    "\n",
    "- We introduce two new quantities:  \n",
    "\n",
    "$$P_j = Q_j U^{-1}_j, \\quad z_j = \\gamma L^{-1}_j e_0.$$\n",
    "\n",
    "- Now we have the following equation for $x_j$:\n",
    "\n",
    "$$ x_j = P_j z_j$$\n",
    "\n",
    "- Due to the recurrence relations, we have\n",
    "\n",
    "$$P_j = \\begin{bmatrix} P_{j-1} & p_j \\end{bmatrix}, $$\n",
    "\n",
    "and \n",
    "\n",
    "$$z_j = \\begin{bmatrix} z_{j-1} \\\\ \\xi_{j} \\end{bmatrix}.$$\n",
    "\n",
    "- For $p_j$ and $\\xi_j$ we have short-term recurrence relations (due to bidiagonal structure)\n",
    "\n",
    "$$p_j = \\frac{1}{d_j}\\left(q_j - b_j p_{j-1} \\right), \\quad \\xi_j = -c_j \\xi_{j-1}.$$\n",
    "\n",
    "- Thus, we arrive at short-term recurrence for $x_j$:\n",
    "\n",
    "$$x_j = P_j z_j = P_{j-1} z_{j-1} + \\xi_j p_j = x_{j-1} + \\xi_j p_j.$$\n",
    "\n",
    "and $q_j$ are found from the Lanczos relation (see slides above).\n",
    "\n",
    "- This method for solving linear systems is called a **direct Lanczos method**. It is closely related to the conjugate gradient method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Direct Lanczos method\n",
    "\n",
    "We have the direct Lanczos method, where we store \n",
    "\n",
    "$$p_{j-1}, q_j, x_{j-1}$$ to get a new estimate of $x_j$.\n",
    "\n",
    "The main problem is with $q_j$: we have the three-term recurrence, but in the floating point arithmetic \n",
    "the orthogonality is can be lost, leading to numerical errors.\n",
    "\n",
    "Let us do some demo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.9605915654183865\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import scipy as sp\n",
    "import scipy.sparse as spsp\n",
    "from scipy.sparse import csc_matrix\n",
    "\n",
    "n = 128\n",
    "ex = np.ones(n);\n",
    "A = spsp.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "rhs = np.ones(n)\n",
    "\n",
    "nit = 64\n",
    "q1 = rhs/np.linalg.norm(rhs)\n",
    "q2 = A.dot(q1)\n",
    "q2 = q2 - np.dot(q2, q1)*q1\n",
    "q2 = q2/np.linalg.norm(q2)\n",
    "qall = [q1, q2]\n",
    "for i in range(nit):\n",
    "    qnew = A.dot(qall[-1])\n",
    "    qnew = qnew - np.dot(qnew, qall[-1])*qall[-1]\n",
    "    qnew = qnew/np.linalg.norm(qnew)\n",
    "    qnew = qnew - np.dot(qnew, qall[-2])*qall[-2]\n",
    "    qnew = qnew/np.linalg.norm(qnew)\n",
    "    qall.append(qnew)\n",
    "qall_mat = np.vstack(qall).T\n",
    "print(np.linalg.norm(qall_mat.T.dot(qall_mat) - np.eye(qall_mat.shape[1])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Conjugate gradient method\n",
    "\n",
    "Instead of $q_j$ (last vector in the modified Gram-Schmidt process), it is more convenient to work with the **residual**\n",
    "\n",
    "$$r_j = f - A x_j.$$\n",
    "\n",
    "The resulting recurrency has the form\n",
    "\n",
    "$x_j = x_{j-1} + \\alpha_{j-1} p_{j-1}$\n",
    "\n",
    "$r_j = r_{j-1} - \\alpha_{j-1}  A p_{j-1}$\n",
    "\n",
    "$p_j = r_j + \\beta_j p_{j-1}$.\n",
    "\n",
    "Hence the name conjugate gradient: to the gradient $r_j$ we add a **conjugate direction** $p_j$.\n",
    "\n",
    "We have **orthogonality** of residuals (check!):\n",
    "\n",
    "$$(r_i, r_j) = 0, \\quad i \\ne j$$\n",
    "\n",
    "and **A-orthogonality** of conjugate directions (check!):\n",
    "\n",
    "$$ (A p_i, p_j) = 0,$$\n",
    "\n",
    "which can be checked from the definition.\n",
    "\n",
    "The equations for $\\alpha_j$ and $\\beta_j$ can be now defined explicitly from these two properties."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## CG final formulas\n",
    "\n",
    "We have $(r_{j}, r_{j-1}) = 0 = (r_{j-1} - \\alpha_{j-1} A p_{j-1}, r_{j-1})$, \n",
    "\n",
    "thus\n",
    "\n",
    "$$\\alpha_{j-1} = \\frac{(r_{j-1}, r_{j-1})}{(A p_{j-1}, r_{j-1})} = \\frac{(r_{j-1}, r_{j-1})}{(A p_{j-1}, p_{j-1} - \\beta_{j-1}p_{j-2})} = \\frac{(r_{j-1}, r_{j-1})}{(A p_{j-1}, p_{j-1})}.$$\n",
    "\n",
    "In the similar way, we have\n",
    "\n",
    "$$\\beta_{j-1} = \\frac{(r_j, r_j)}{(r_{j-1}, r_{j-1})}.$$\n",
    "\n",
    "Recall that\n",
    "\n",
    "$x_j = x_{j-1} + \\alpha_{j-1} p_{j-1}$\n",
    "\n",
    "$r_j = r_{j-1} - \\alpha_{j-1}  A p_{j-1}$\n",
    "\n",
    "$p_j = r_j + \\beta_j p_{j-1}$.\n",
    "\n",
    "Only one matrix-by-vector product per iteration.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## CG derivation overview\n",
    "\n",
    "- Want to find $x_*$ in Krylov subspace\n",
    "- But natural basis is ill-conditioned, therefore we need orthogonalization\n",
    "- Derive recurrent equation for sequential orthogonalization of the Krylov subspace basis\n",
    "    - Arnoldi process for non-symmetric matrix\n",
    "    - Lanczos process for symmetrix matrix\n",
    "- Clever re-writing of these formulas gives short recurrence "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Some history\n",
    "More details here: https://www.siam.org/meetings/la09/talks/oleary.pdf\n",
    "\n",
    "When Hestenes worked on conjugate bases in 1936, he was advised by a\n",
    "Harvard professor that it was too obvious for publication\n",
    "- CG doesn’t work on slide rules.\n",
    "<img src=\"slide_rule.jpg\">\n",
    "- CG has little advantage over Gauss elimination for computation with\n",
    "calculators.\n",
    "- CG is not well suited for a room of human \"computers\" – too much data exchange.\n",
    "<img src=\"human_computers.jpg\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Properties of the CG method\n",
    "\n",
    "- We need to store 3 vectors.\n",
    "\n",
    "- Since it generates $A$-orthogonal sequence $p_1, \\ldots, p_N$, after $n$ steps it should stop (i.e., $p_{N+1} = 0$.) \n",
    "\n",
    "- In practice it does not have this property in finite precision, thus after its invention in 1952 by Hestens and Stiefel it was labeled **unstable**.\n",
    "\n",
    "- In fact, it is a brilliant iterative method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## $A$-optimality\n",
    "\n",
    "\n",
    "Energy functional can be written as \n",
    "\n",
    "$$(Ax, x) - 2(f, x) = (A (x - x_*), (x - x_*)) - (Ax _*, x_*),$$\n",
    "\n",
    "where $A x_* = f$. Up to a constant factor,\n",
    "\n",
    "$$ (A(x - x_*), (x -x_*)) = \\Vert x - x_* \\Vert^2_A$$\n",
    "\n",
    "is the **A-norm** of the error.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Convergence\n",
    "The CG method computes $x_k$ that minimizes the energy functional over the Krylov subspace, i.e. $x_k = p(A)f$, where $p$ is a polynomial of degree $k+1$, so\n",
    "\n",
    "$$\\Vert x_k - x_* \\Vert_A  =  \\inf\\limits_{p} \\Vert \\left(p(A) - A^{-1}\\right) f \\Vert_A. $$\n",
    "\n",
    "Using eigendecomposition of $A$ we have\n",
    "\n",
    "$$A = U \\Lambda U^*, \\quad  g = U^* f,$$ and\n",
    "\n",
    "$\\Vert x - x_* \\Vert^2_A = \\displaystyle{\\inf_p} \\Vert \\left(p(\\Lambda) - \\Lambda^{-1}\\right) g \\Vert_\\Lambda^2 = \\displaystyle{\\inf_p}\n",
    "\\displaystyle{\\sum_{i=1}^n} \\frac{(\\lambda_i p(\\lambda_i) - 1)^2 g^2_i}{\\lambda_i} = \\displaystyle{\\inf_{q, q(0) = 1}} \\displaystyle{\\sum_{i=1}^n} \\frac{q(\\lambda_i)^2 g^2_i}{\\lambda_i}\n",
    "$\n",
    "\n",
    "Selection of the optimal $q$ depends on the eigenvalue distribution.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Absolute and relative error\n",
    "\n",
    "We have\n",
    "\n",
    "$$\\Vert x - x_* \\Vert^2_A \\leq \\sum_{i=1}^n \\frac{g^2_i}{\\lambda_i} \\inf_{q, q(0)=1} \\max_{j} q({\\lambda_j})^2$$\n",
    "\n",
    "The first term is just \n",
    "\n",
    "$$\\sum_{i=1}^n \\frac{g^2_i}{\\lambda_i} = (A^{-1} f, f) = \\Vert x_* \\Vert^2_A.$$\n",
    "\n",
    "And we have relative error bound\n",
    "\n",
    "$$\\frac{\\Vert x - x_* \\Vert_A }{\\Vert x_* \\Vert_A} \\leq \\inf_{q, q(0)=1} \\max_{j} |q({\\lambda_j})|,$$\n",
    "\n",
    "so if matrix has only 2 different eigenvalues, then there exists a polynomial of degree 2 such that $q({\\lambda_1}) =q({\\lambda_2})=0$, so in this case CG converges in 2 iterations.\n",
    "\n",
    "<font color='red'>\n",
    "\n",
    "- If eigenvalues are clustered and there are $l$ outliers, then after first $\\mathcal{O}(l)$ iterations CG will converge as if there are no outliers (and hence the effective condition number is smaller).\n",
    "    \n",
    "- The intuition behind this fact is that after $\\mathcal{O}(l)$ iterations the polynomial has degree more than $l$ and thus is able to zero $l$ outliers.\n",
    "</font>\n",
    "\n",
    "Let us find another useful upper-bound estimate of convergence. \n",
    "Since\n",
    "\n",
    "$$\n",
    "\\inf_{q, q(0)=1} \\max_{j} |q({\\lambda_j})| \\leq \\inf_{q, q(0)=1} \\max_{\\lambda\\in[\\lambda_\\min,\\lambda_\\max]} |q({\\lambda})|\n",
    "$$\n",
    "\n",
    "The last term is just the same as for the Chebyshev acceleration, thus the same \n",
    "upper convergence bound holds:\n",
    "\n",
    "$$\\frac{\\Vert x_k - x_* \\Vert_A }{\\Vert x_* \\Vert_A} \\leq \\gamma \\left( \\frac{\\sqrt{\\mathrm{cond}(A)}-1}{\\sqrt{\\mathrm{cond}(A)}+1}\\right)^k.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Finite termination & clusters\n",
    "\n",
    "1. If $A$ has only $m$ different eigenvalues, CG converges in $m$ iterations (proof in the blackboard).\n",
    "2. If $A$ has $m$ \"clusters\" of eigenvalues, CG converges cluster-by-cluster.\n",
    "\n",
    "As a result, better convergence than Chebyshev acceleration, but slightly higher cost per iteration."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Summary\n",
    "\n",
    "CG is the method of choice for symmetric positive definite systems:\n",
    "\n",
    "1. $\\mathcal{O}(n)$ memory\n",
    "2. Square root of condition number in the estimates\n",
    "3. Automatic ignoring of the outliers/clusters\n",
    "4. $A$-optimality property"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Non-linear conjugate gradient method\n",
    "\n",
    "- CG minimizes the energy functional, which is quadratic in $x$\n",
    "- CG formulas were used as starting point in developing methods to minimize arbitrary convex function\n",
    "- Most popular CG extensions (so-called non-linear CG method) are \n",
    "    - [Hestenes-Stiefel method](https://www.fing.edu.uy/inco/cursos/numerico/aln/hes_stief1952.pdf)\n",
    "    - [Polak-Ribiere method](http://www.numdam.org/article/M2AN_1969__3_1_35_0.pdf) - original paper in French\n",
    "    - [Fletcher–Reeves method](https://watermark.silverchair.com/070149.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAkMwggI_BgkqhkiG9w0BBwagggIwMIICLAIBADCCAiUGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMiWM0crDA8bZ0Fj5HAgEQgIIB9tIM8xbxhkhniJQifurRLHIaOcnos9gvgBj8Lkc34eBUtUg4oYzjusGP_FZiHfpBoRnM409FFCs-Y5xxomumnGRQ8UIZJe7CPSmaCcSklAdwHN3Ja-fHfALctP68U3mGGeR11P8vlESoS7HZhZEmVhFb5m7jPk1ENZKLlq9CnVTz2n2bI3HPYCfDIo8ywd8RWo5gw6EwYnZ46l1ExES8AHMqlklpgzdupLhL6HxTzBEHd_qY4B2FjK4m4MpvdaX5imEnOD6YCwZMNC_qLzUdPxumcXl4oTRDyJp1kg_6PcGi98hE10InOp4aHJeIjl5FPJnuEFEBcv28asOq4vngp6N5d-4y1fpeoxcAe4d857TeLtQnGD6LlCAL_3Mh0lQ8iYLLfjw4b_V2jHEuSxKH3BalLPMS5KF2nGqqrzfbDs8x9jW-OlkeL_x68Zidv0rTHqTd0ww3kvSINDjgw6owPMLK9KxEWP5T-4dnu_kn-NXR1W0s52kUHFD1U9AWs5Zzh4RTVOXsdjGQciWzOO-XieLn8Aju-0P8mVEGjUDOjLDZtuPYM-Ep_Z9lp8TNDCi2lB5IQFbKVpBFaAItDDyVUvKNPfBovmKlnPRFLZYjLbgocnU3tcy0YB8RZegC56Gxn7g_WkphptCVPm7IDeCE9l5gcFafuY0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Non-symmetric systems and the generalized minimal residual method (GMRES) [(Y. Saad, M. Schultz, 1986)](https://epubs.siam.org/doi/10.1137/0907058)\n",
    "\n",
    "Before we discussed symmetric positive definite systems. What happens if $A$ is non-symmetric?\n",
    "\n",
    "We can still orthogonalize the Krylov subspace using Arnoldi process, and get\n",
    "\n",
    "$$A Q_j = Q_j H_j + h_{j,j-1}q_j e^{\\top}_{j-1}.$$\n",
    "\n",
    "Let us rewrite the latter expression as\n",
    "\n",
    "$$ A Q_j = Q_j H_j + h_{j,j-1}q_j e^{\\top}_{j-1} = Q_{j+1} \\widetilde H_j, \\quad \\widetilde H_j = \n",
    "\\begin{bmatrix} h_{0,0} & h_{0,1} & \\dots & h_{0,j-2} & h_{0,j-1} \\\\ h_{1,0} & h_{1,1} & \\dots & h_{1,j-2} & h_{1,j-1} \\\\ 0& h_{2,2} &  \\dots & h_{2,j-2} & h_{2,j-1} \\\\\n",
    "0& 0 & \\ddots & \\vdots & \\vdots  \\\\\n",
    "0& 0 &  & h_{j,j-1} & h_{j-1,j-1} \\\\ 0& 0 & \\dots & 0 & h_{j,j-1}\\end{bmatrix}$$\n",
    "\n",
    "Then, if we need to minimize the residual over the Krylov subspace, we have\n",
    "\n",
    "$$x_j = x_0 + Q_j c_j $$\n",
    "\n",
    "and $x_j$ has to be selected as \n",
    "\n",
    "$$ \\Vert A x_j - f \\Vert_2 =  \\Vert A Q_j c_j - r_0 \\Vert_2 \\rightarrow \\min_{c_j}.$$\n",
    "\n",
    "Using the Arnoldi recursion, we have\n",
    "\n",
    "$$ \\Vert Q_{j+1} \\widetilde H_j c_j -  r_0 \\Vert_2 \\rightarrow \\min_{c_j}.$$\n",
    "\n",
    "Using the orthogonal invariance under multiplication by unitary matrix, we get\n",
    "\n",
    "$$ \\Vert \\widetilde H_j c_j - \\gamma e_0 \\Vert_2 \\rightarrow \\min_{c_j},$$\n",
    "\n",
    "where we have used that $Q^*_{j+1} r_0 = \\gamma e_0, \\gamma = \\Vert r_0 \\Vert$\n",
    "\n",
    "- This is just a linear least squares with $(j+1)$ equations and $j$ unknowns.\n",
    "\n",
    "- The matrix is also upper Hessenberg, thus its QR factorization can be computed in a very cheap way.\n",
    "\n",
    "- This allows the computation of $c_j$. This method is called **GMRES** (generalized minimal residual)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Summary of the GMRES\n",
    "\n",
    "- Minimizes the residual directly\n",
    "- No normal equations\n",
    "- Memory grows with the number of iterations as $\\mathcal{O}(nj)$, so **restarts** typically implemented (just start GMRES from the new initial guess)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 1000x500 with 0 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1000x500 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.sparse.linalg as la\n",
    "from scipy.sparse import csc_matrix, csr_matrix\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import time\n",
    "\n",
    "%matplotlib inline\n",
    "n = 150\n",
    "ex = np.ones(n);\n",
    "lp1 = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "e = sp.sparse.eye(n)\n",
    "A = sp.sparse.kron(lp1, e) + sp.sparse.kron(e, lp1)\n",
    "A = csr_matrix(A)\n",
    "rhs = np.ones(n * n)\n",
    "\n",
    "plt.figure(figsize=(10, 5))\n",
    "f, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))\n",
    "for restart in [5, 40, 200]:\n",
    "    hist = []\n",
    "    def callback(rk):\n",
    "        hist.append(np.linalg.norm(rk) / np.linalg.norm(rhs))\n",
    "    st = time.time()\n",
    "    sol = la.gmres(A, rhs, x0=np.zeros(n*n), maxiter=200, restart=restart, callback=callback, tol=1e-16)\n",
    "    current_time = time.time() - st\n",
    "    ax1.semilogy(np.array(hist), label='rst={}'.format(restart))\n",
    "    ax2.semilogy([current_time * i / len(hist) for i in range(len(hist))], np.array(hist), label='rst={}'.format(restart))\n",
    "    \n",
    "\n",
    "ax1.legend(loc='best')\n",
    "ax2.legend(loc='best')\n",
    "ax1.set_xlabel(\"Number of outer iterations\", fontsize=20)\n",
    "ax2.set_xlabel(\"Time, sec\", fontsize=20)\n",
    "ax1.set_ylabel(r\"$\\frac{||r_k||_2}{||rhs||_2}$\", fontsize=20)\n",
    "ax2.set_ylabel(r\"$\\frac{||r_k||_2}{||rhs||_2}$\", fontsize=20)\n",
    "plt.sca(ax1)\n",
    "plt.yticks(fontsize=20)\n",
    "plt.sca(ax2)\n",
    "plt.yticks(fontsize=20)\n",
    "f.tight_layout()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.sparse.linalg as la\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# Example from http://www.caam.rice.edu/~embree/39961.pdf\n",
    "\n",
    "A = np.array([[1, 1, 1],\n",
    "              [0, 1, 3],\n",
    "              [0, 0, 1]]\n",
    "            )\n",
    "rhs = np.array([2, -4, 1])\n",
    "x0 = np.zeros(3)\n",
    "\n",
    "for restart in [1, 2, 3]:\n",
    "    hist = []\n",
    "    def callback(rk):\n",
    "        hist.append(np.linalg.norm(rk)/np.linalg.norm(rhs))\n",
    "    _ = la.gmres(A, rhs, x0=x0, maxiter=20, restart=restart, callback=callback)\n",
    "    plt.semilogy(np.array(hist), label='rst={}'.format(restart))\n",
    "plt.legend(fontsize=22)\n",
    "plt.xlabel(\"Number of outer iterations\", fontsize=20)\n",
    "plt.ylabel(r\"$\\frac{||r_k||_2}{||rhs||_2}$\", fontsize=20)\n",
    "plt.xticks(fontsize=16)\n",
    "plt.yticks(fontsize=20)\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Next lecture\n",
    "\n",
    "- Iterative methods continued (BiCG, MINRES)\n",
    "- Preconditioners"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Questions?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
       "<style>\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "    }\n",
       "    div.cell{\n",
       "        /*width:80%;*/\n",
       "        /*margin-left:auto !important;\n",
       "        margin-right:auto;*/\n",
       "    }\n",
       "    h1 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
       "    }\n",
       "    h2 {\n",
       "        font-family: 'Fenix', serif;\n",
       "    }\n",
       "    h3{\n",
       "\t\tfont-family: 'Fenix', serif;\n",
       "        margin-top:12px;\n",
       "        margin-bottom: 3px;\n",
       "       }\n",
       "\th4{\n",
       "\t\tfont-family: 'Fenix', serif;\n",
       "       }\n",
       "    h5 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
       "    }\t   \n",
       "    div.text_cell_render{\n",
       "        font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "        line-height: 1.2;\n",
       "        font-size: 120%;\n",
       "        /*width:70%;*/\n",
       "        /*margin-left:auto;*/\n",
       "        margin-right:auto;\n",
       "    }\n",
       "    .CodeMirror{\n",
       "            font-family: \"Source Code Pro\";\n",
       "\t\t\tfont-size: 90%;\n",
       "    }\n",
       "/*    .prompt{\n",
       "        display: None;\n",
       "    }*/\n",
       "    .text_cell_render h1 {\n",
       "        font-weight: 200;\n",
       "        font-size: 50pt;\n",
       "\t\tline-height: 110%;\n",
       "        color:#CD2305;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\t\n",
       "    .text_cell_render h5 {\n",
       "        font-weight: 300;\n",
       "        font-size: 16pt;\n",
       "        color: #CD2305;\n",
       "        font-style: italic;\n",
       "        margin-bottom: .5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\n",
       "    \n",
       "    li {\n",
       "        line-height: 110%;\n",
       "    }\n",
       "    .warning{\n",
       "        color: rgb( 240, 20, 20 )\n",
       "        }  \n",
       "\n",
       "</style>\n",
       "\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"]\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = open(\"./styles/custom.css\", \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  },
  "nav_menu": {},
  "rise": {
   "scroll": true,
   "theme": "sky",
   "transition": "zoom"
  },
  "toc": {
   "navigate_menu": true,
   "number_sections": false,
   "sideBar": true,
   "threshold": 6,
   "toc_cell": false,
   "toc_section_display": "block",
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}