gram-schmidt-functions.ipynb

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     "text": [
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/srv/rep/22-course-cse6643/mybinder/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/srv/rep/22-course-cse6643/mybinder/Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "import Pkg; Pkg.add([\"QuadGK\", \"Plots\"]);\n",
    "using LinearAlgebra;\n",
    "using Plots;default(l=2, size=(900,550), label=false);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "554af06a",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Gram-Schmidt Orthogonalization on Vectors that are Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "926caca0",
   "metadata": {
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    "slideshow": {
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   },
   "source": [
    "## The inputs / outputs to our implementations of QR factorization have been matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce955445",
   "metadata": {},
   "source": [
    "Like this implementation of modified Gram-Schmidt:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f27524ca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mgs (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mgs(A)\n",
    "    _, n = size(A)\n",
    "    Q = copy(A)\n",
    "    R = zeros(eltype(A), n, n)\n",
    "    for i in 1:n\n",
    "        R[i,i] = norm(Q[:,i])\n",
    "        Q[:,i] /= R[i,i]\n",
    "        R[i,i+1:n] = Q[:,i]' * Q[:,i+1:n]\n",
    "        Q[:,i+1:n] -= Q[:,[i]] * R[[i],i+1:n]\n",
    "    end \n",
    "    return Q, R\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a945e05",
   "metadata": {
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   },
   "source": [
    "## But $A$ and $Q$ can be treated as arrays of column vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6cdecd2",
   "metadata": {},
   "source": [
    "Like this implementation of modified Gram-Schmidt:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "289aa10c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mgs_vectors (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
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   ],
   "source": [
    "function mgs_vectors(A::Array{V,1}) where V # V is the type of the column vector\n",
    "    n = length(A) # the number of vectors\n",
    "    Q = collect(copy.(A)) # copy all of the individual vectors\n",
    "    R = zeros(eltype(V), n, n) \n",
    "    for i in 1:n\n",
    "        R[i,i] = sqrt(dot(Q[i], Q[i])) # definition of norm(Q[i])\n",
    "        Q[i] /= R[i,i]\n",
    "        for j in i+1:n\n",
    "            R[i,j] = dot(Q[i], Q[j])\n",
    "            Q[j] -= Q[i] * R[i,j]\n",
    "        end\n",
    "    end\n",
    "    return Q, R\n",
    "end"
   ]
  },
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   "id": "d5e99c91",
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   "source": [
    "### A demonstration that these are the same algorithm"
   ]
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   "id": "ff8b71d4",
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   "source": [
    "m = 6; n = 4;\n",
    "A = randn(4,4);\n",
    "A_vecs = collect(A[:,i] for i in 1:n);"
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
   "id": "3ce55b54",
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   "source": [
    "Q, R = mgs(A);"
   ]
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   "id": "eb56a9e2",
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   "source": [
    "Q_vecs, R_vecs = mgs_vectors(A_vecs);"
   ]
  },
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   "id": "a2ea0576",
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       "2.9893669801409083e-16"
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   "source": [
    "norm(R - R_vecs)"
   ]
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   "source": [
    "maximum(norm(Q[:,i] - Q_vecs[i]) for i in 1:n)"
   ]
  },
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   "id": "1d43fb58",
   "metadata": {
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   },
   "source": [
    "## What does `mgs_vectors` need the type `V` to provide? What doesn't it need?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfa20587",
   "metadata": {
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   "source": [
    "**Does need:**\n",
    "\n",
    "1. number type `eltype(V)`\n",
    "2. vector addition `v + w`\n",
    "3. vector scaling `a * v`\n",
    "4. inner product `dot(v, w)`"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bac04aa3",
   "metadata": {
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   "source": [
    "**Does not need:**\n",
    "\n",
    "1. The dimension of the space `length(v)`\n",
    "2. Access to individual coefficients `v[i]`"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8408c934",
   "metadata": {},
   "source": [
    "- The things it _does need_ describe a vector space with an inner product.\n",
    "- The things it _does not need_ are specific to $\\mathbb{C}^m$ or $\\mathbb{R}^m$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f63529a",
   "metadata": {
    "slideshow": {
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    }
   },
   "source": [
    "## Recall that functions on an interval form a vector space\n",
    "\n",
    "- We mentioned this in a lecture, but let's repeat:\n",
    "\n",
    "> The set $V[a,b] = \\{f: [a, b] \\to \\mathbb{C}\\}$ is a vector space, with vector addition being the function $(f + g): x \\mapsto f(x) + g(x)$ and vector scaling $(\\alpha f): x \\mapsto \\alpha f(x).$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "350a5d96",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Define a type for functions as vectors\n",
    "\n",
    "- Let's give ourselves our own type for functions on an interval.  To emphasize that they are vectors, our type will use indexing notation to evaluate the function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "480ee091",
   "metadata": {
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   "source": [
    "# Define type Func for function taking values in field F\n",
    "struct Func{F <: Number}\n",
    "    _f # inner callable\n",
    "end\n",
    "\n",
    "Base.copy(f::Func{F}) where F = f\n",
    "Base.eltype(::Type{<:Func{F}}) where F = F\n",
    "# evaluate at one point\n",
    "Base.getindex(f::Func{F},x) where F where T = F(f._f(x))\n",
    "# evaluate at many points\n",
    "Base.getindex(f::Func{F},v::AbstractVector{T}) where F where T = collect(F(f._f(x)) for x in v)"
   ]
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   "source": [
    "Our type can be evaluated at points, but notice that we can't get its length and we can't get `v[:]` AKA \"all the components\""
   ]
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   "id": "3ba2266b",
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   "source": [
    "f = Func{Float64}(x->x^2);"
   ]
  },
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   "id": "b3e9ce46",
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    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 1.0\n",
       " 4.0\n",
       " 9.0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f[[1,2,3]]"
   ]
  },
  {
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   "execution_count": 12,
   "id": "013fbe11",
   "metadata": {
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   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "MethodError: no method matching length(::Func{Float64})\n\u001b[0mClosest candidates are:\n\u001b[0m  length(\u001b[91m::Union{Base.KeySet, Base.ValueIterator}\u001b[39m) at ~/srv/rep/julia/usr/share/julia/base/abstractdict.jl:58\n\u001b[0m  length(\u001b[91m::Union{ZMQ._Message, Base.RefValue{ZMQ._Message}}\u001b[39m) at ~/.julia/packages/ZMQ/R3wSD/src/_message.jl:31\n\u001b[0m  length(\u001b[91m::Union{DataStructures.OrderedRobinDict, DataStructures.RobinDict}\u001b[39m) at ~/.julia/packages/DataStructures/vSp4s/src/ordered_robin_dict.jl:86\n\u001b[0m  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching length(::Func{Float64})\n\u001b[0mClosest candidates are:\n\u001b[0m  length(\u001b[91m::Union{Base.KeySet, Base.ValueIterator}\u001b[39m) at ~/srv/rep/julia/usr/share/julia/base/abstractdict.jl:58\n\u001b[0m  length(\u001b[91m::Union{ZMQ._Message, Base.RefValue{ZMQ._Message}}\u001b[39m) at ~/.julia/packages/ZMQ/R3wSD/src/_message.jl:31\n\u001b[0m  length(\u001b[91m::Union{DataStructures.OrderedRobinDict, DataStructures.RobinDict}\u001b[39m) at ~/.julia/packages/DataStructures/vSp4s/src/ordered_robin_dict.jl:86\n\u001b[0m  ...",
      "",
      "Stacktrace:",
      " [1] top-level scope",
      "   @ In[12]:1",
      " [2] eval",
      "   @ ./boot.jl:373 [inlined]",
      " [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
      "   @ Base ./loading.jl:1196"
     ]
    }
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   "source": [
    "length(f)"
   ]
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   "id": "aee50a2a",
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    {
     "ename": "LoadError",
     "evalue": "MethodError: no method matching ^(::Colon, ::Int64)\n\u001b[0mClosest candidates are:\n\u001b[0m  ^(\u001b[91m::Union{AbstractChar, AbstractString}\u001b[39m, ::Integer) at ~/srv/rep/julia/usr/share/julia/base/strings/basic.jl:721\n\u001b[0m  ^(\u001b[91m::UniformScaling\u001b[39m, ::Number) at ~/srv/rep/julia/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/uniformscaling.jl:318\n\u001b[0m  ^(\u001b[91m::Diagonal\u001b[39m, ::Integer) at ~/srv/rep/julia/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/diagonal.jl:196\n\u001b[0m  ...",
     "output_type": "error",
     "traceback": [
      "MethodError: no method matching ^(::Colon, ::Int64)\n\u001b[0mClosest candidates are:\n\u001b[0m  ^(\u001b[91m::Union{AbstractChar, AbstractString}\u001b[39m, ::Integer) at ~/srv/rep/julia/usr/share/julia/base/strings/basic.jl:721\n\u001b[0m  ^(\u001b[91m::UniformScaling\u001b[39m, ::Number) at ~/srv/rep/julia/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/uniformscaling.jl:318\n\u001b[0m  ^(\u001b[91m::Diagonal\u001b[39m, ::Integer) at ~/srv/rep/julia/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/diagonal.jl:196\n\u001b[0m  ...",
      "",
      "Stacktrace:",
      " [1] literal_pow",
      "   @ ./intfuncs.jl:325 [inlined]",
      " [2] (::var\"#7#8\")(x::Function)",
      "   @ Main ./In[10]:1",
      " [3] getindex(f::Func{Float64}, x::Function)",
      "   @ Main ./In[9]:9",
      " [4] top-level scope",
      "   @ In[13]:1",
      " [5] eval",
      "   @ ./boot.jl:373 [inlined]",
      " [6] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
      "   @ Base ./loading.jl:1196"
     ]
    }
   ],
   "source": [
    "f[:] "
   ]
  },
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   "id": "80cb061a",
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   "source": [
    "## Implement the vector algebra rules for functions\n",
    "\n",
    "> The set $V[a,b] = \\{f: [a, b] \\to \\mathbb{C}\\}$ is a vector space, with vector addition being the function $(f + g): x \\mapsto f(x) + g(x)$ and vector scaling $(\\alpha f): x \\mapsto \\alpha f(x).$"
   ]
  },
  {
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   "execution_count": 14,
   "id": "46787c6f",
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   "source": [
    "# import these operators so we can extend them\n",
    "import Base: *,/,\\,+,-\n",
    "\n",
    "# Define scaling\n",
    "*(a::N, v::Func{F}) where N <: Number where F = Func{F}(x -> a * v._f(x))\n",
    "# Define addition\n",
    "+(v::Func{F}, w::Func{F}) where F = Func{F}(x -> v._f(x) + w._f(x));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d7c8c45f",
   "metadata": {
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   "source": [
    "### With some syntactic sugar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "9ddfc76c",
   "metadata": {
    "cell_style": "center"
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   "outputs": [],
   "source": [
    "# multiplication on the right = multiplication on the left\n",
    "*(v::Func{F}, a::N) where F where N <: Number = a * v\n",
    "# v / a = (1/a) * v\n",
    "/(v::Func{F}, a::N) where F where N <: Number = (1/a) * v\n",
    "# v - w = v + (-1 * w)\n",
    "-(v::Func{F}, w::Func{F}) where F = v + ((-1) * w);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43cfb1aa",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### In action"
   ]
  },
  {
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   "execution_count": 16,
   "id": "b2f00d72",
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    "slideshow": {
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   "outputs": [
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      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = Func{Float64}(x -> -x)\n",
    "w = Func{Float64}(x -> x^2)\n",
    "vpw = v + w\n",
    "sw = -0.5 * w\n",
    "x = LinRange(0, 1, 200)\n",
    "\n",
    "plot(x, [v[x] w[x] vpw[x] sw[x]], labels = [\"v\" \"w\" \"v+w\" \"(-1/2)w\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7cb450d4",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## We can define inner products on vectors spaces other than $\\mathbb{C}^m$, including on functions on an interval\n",
    "\n",
    "- Recall that an inner product has axioms, just like a vector space.  For a complex vector space $V$, if I can define an operation $(\\cdot,\\cdot):(V \\otimes V) \\to \\mathbb{C}$ that satisfies the inner product axioms, then $(\\cdot,\\cdot)$ is an inner product.  Those axioms again are:\n",
    "\n",
    "1. $(u, v) = \\overline{(v, u)}$ (conjugate symmetry)\n",
    "2. $(u, \\alpha v) = \\alpha(u, v)$ and $(u, v + w) = (u, v) + (u, w)$ (linearity in the second argument, which combined with property 1 implies conjugate linearity in the first argument)\n",
    "3. $(u, u) \\geq 0$ with $[(u, u) = 0] \\Leftrightarrow [u = 0]$ (positive definiteness)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "792a67ba",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## The simplest inner product on functions is the $L^2$ inner product\n",
    "\n",
    "- Consider just continuous functions on the interval $[a,b]$,\n",
    "\n",
    "$$C^0[a,b] = \\{f:[a,b] \\to \\mathbb{C}, \\text{such that } f \\text{ is continuous}\\}.$$\n",
    "\n",
    "- Define the operation $(\\cdot,\\cdot): (C^0[a,b] \\otimes C^0[a,b]) \\to \\mathbb{C}$ by\n",
    "$$(f, g) = \\int_a^b \\overline{f(x)} g(x)\\ dx.$$\n",
    "\n",
    "- This is the **$L^2$ inner product**.  Verifying the inner product axioms is an exercise you should try if you're not convinced.\n",
    "\n",
    "- _Every_ inner product defines a norm, so there is an $L^2$ norm on $C^0[a,b]$,\n",
    "\n",
    "$$\\|f\\|_{L^2} = \\left\\{\\int_a^b |f(x)|^2\\ dx\\right\\}^{1/2}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab41bcf0",
   "metadata": {
    "hideCode": true,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Let's (approximately) implement the $L^2$ inner product\n",
    "\n",
    "- Methods that evaluate $f(x)$ at a finite number of points in order to estimate $\\int f(x)\\ dx$ are called _quadrature_ rules.  Let's get some numerical quadrature methods from the `QuadGK` package.  Example usage:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "40dae13d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.0, 1.0428675273326288e-9)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using QuadGK\n",
    "\n",
    "integral, err = quadgk(x -> exp(-x^2 / 2) / sqrt(2 * π), -10, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c703bc5a",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Equations to code\n",
    "\n",
    "$$(f, g) = \\int_a^b \\overline{f(x)} g(x)\\ dx.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "ab3b0ae2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "L2_inner_product (generic function with 1 method)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function L2_inner_product(f, g; a=-1, b=1, rtol=1.e-12)\n",
    "    val, _ = quadgk(x -> conj(f(x)) * g(x), a, b,  rtol=rtol)\n",
    "    return val\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "46b3e96c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.39999999999999997"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L2_inner_product(x -> -x, x-> x^3) # ∫₋₁,₁ (-x)*x³ dx = -∫₋₁,₁ x⁴ dx = -(1/5)(1⁵ - (-1)⁵) = -1/5 * 2 = -2/5"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2fcd254",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Define the $L^2$ inner product to be `dot` for our function type"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f468ae13",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dot (generic function with 74 methods)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import LinearAlgebra: dot\n",
    "\n",
    "dot(v::Func{F}, w::Func{F}) where F = L2_inner_product(v._f, w._f)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a29b2d42",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Basis functions with large cosine similarity have problems"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31a44d1b",
   "metadata": {},
   "source": [
    "- The angle between the two angles on the left is $1$ degree: how well can you tell them apart?\n",
    "- The angle (defined as $(f,g) / (\\|f\\|\\ \\|g\\|)$) between $x^{55}$ and $x^{57}$ is about $1$ degree: how well can you tell them apart?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "446228b0",
   "metadata": {
    "hideCode": true
   },
   "outputs": [
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      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = [0.5,0.5]\n",
    "θ = π / 180\n",
    "c, s = [cos(θ), sin(θ)]\n",
    "w = [[c,s] [-s,c]] * v\n",
    "p = plot([[0,v[1]] [0,w[1]]], [[0,v[2]] [0,w[2]]]\n",
    "    , aspect_ratio = :equal\n",
    "    , lim=(-.8,.8)\n",
    "    , framestyle = :origin\n",
    "    )\n",
    "\n",
    "f = Func{Float64}(x->x^55)\n",
    "g = Func{Float64}(x->x^57)\n",
    "x = LinRange(-1,1,200)\n",
    "q = plot(x, [f[x] g[x]], label=[\"x⁵⁵\" \"x⁵⁷\"], framestyle=:origin)\n",
    "plot(p,q,layout=(1,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3da2c95",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Using Gram-Schmidt to create orthonormal bases"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f760ccc",
   "metadata": {},
   "source": [
    "Suppose we want a good basis of functions for $\\mathcal{P}^5[-1,1]$: polynomials up to degree 5 on the interval $[-1,1]$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "775b17da",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Start with a basis that is easy to define and evaluate: the _monomial basis_\n",
    "\n",
    "The monomial basis is $\\{x^0, x^1, x^2, x^3, x^4, x^5\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "51526a2d",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 5\n",
    "A = collect(Func{Float64}(x->x^i) for i in 0:p);\n",
    "x = LinRange(-1,1,200)\n",
    "plot(x, hcat(collect(A[i][x] for i in 1:p+1)...)\n",
    "    , framestyle=:origin\n",
    "    , label=hcat(collect(\"x^$i\" for i in 0:p)...))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f534d519",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Run Gram-Schmidt to Construct an Orthonormal Basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "e9be06c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "Q, R = mgs_vectors(A);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "ad98cc22",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(x, hcat(collect(Q[i][x] for i in 1:p+1)...)\n",
    "    , framestyle=:origin\n",
    "    , label=hcat(collect(\"q_$i\" for i in 0:p)...))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a652b732",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "These polynomials are scaled versions of the [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61e6977b",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## We can orthogonalize nonpolynomials using Gram-Schmidt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e062434",
   "metadata": {},
   "source": [
    "Here we orthogonalize the set $\\{e^x, \\tanh x, \\sin x, \\text{ReLU} x, e^{-x^2}\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "7a277203",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = map(f -> Func{Float64}(f),[x -> exp(x), x -> tanh(x), x -> sin(x), x -> if (x < 0) 0 else x end, x -> exp(-x^2)]);\n",
    "x = LinRange(-1, 1, 200)\n",
    "plot(x, hcat(collect(B[i][x] for i in 1:5)...), framestyle=:origin, ylim=(-3,3), label=[\"eˣ\" \"tanh x\" \"sin x\" \"ReLU x\" \"exp(-x²)\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "a32d481c",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "Q, R = mgs_vectors(B);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "7b4aecad",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(x, hcat(collect(Q[i][x] for i in 1:5)...), framestyle=:origin, ylim=(-3,3), label=[\"q_1\" \"q_2\" \"q_3\" \"q_4\" \"q_5\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0de21a35",
   "metadata": {
    "slideshow": {
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    }
   },
   "source": [
    "## Notes on HW2 Problem 7\n",
    "\n",
    "Do not use `QuadGK` to approximate the $L^2$ inner product.  You should select $m$ evenly spaced points $\\{x_i\\}_{i=1}^m$ in $[a,b]$, approximate the function $f(x)$ as the vector $\\hat{f}$ of values $[\\hat{f}_1, \\dots, \\hat{f}_m]$ where $\\hat{f}_i = f(x_i)$, and approximate the inner-product as\n",
    "\n",
    "$$(f,g) \\approx \\frac{(b-a)}{m} \\hat{f}^* \\hat{g}.$$"
   ]
  }
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