In [1]:
## Lecture 8, look at quadratic.
A = [3 -1.0; -1.0 4]
b = [-1, 2.0]
xsoln = A\b
Out[1]:
2-element Vector{Float64}:
 -0.1818181818181818
  0.4545454545454546
In [2]:
##
# A quadratic function is
# f(x) = 1/2 x'*A*x - x'*b
# We will show, miniminzing f(x) <=> solving Ax = b when A is symm. pos. def.
# sym. pos. def <=> x'*A*x > 0
quadratic(A,b,x) = 0.5*x'*A*x - x'*b
@show quadratic(A,b,xsoln)
@show quadratic(A,b,randn(2))

quadratic(A, b, xsoln) = -0.5454545454545455
quadratic(A, b, randn(2)) = 7.1035053446817
Out[2]:
7.1035053446817
In [3]:
##
using Plots
pyplot()
x = y = range(-1, stop = 1, length = 25)
plot(x,y,(x,y) -> quadratic(A,b,[x,y]), st=:surface,camera=(80,30))
scatter!([xsoln[1]],[xsoln[2]], [quadratic(A, b, xsoln)], markersize = 12)
gui()

ArgumentError: Package PyPlot not found in current path.
- Run `import Pkg; Pkg.add("PyPlot")` to install the PyPlot package.

Stacktrace:
  [1] macro expansion
    @ ./loading.jl:1772 [inlined]
  [2] macro expansion
    @ ./lock.jl:267 [inlined]
  [3] __require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1753
  [4] #invoke_in_world#3
    @ ./essentials.jl:926 [inlined]
  [5] invoke_in_world
    @ ./essentials.jl:923 [inlined]
  [6] require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1746
  [7] top-level scope
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:883
  [8] eval
    @ ./boot.jl:385 [inlined]
  [9] _initialize_backend(pkg::Plots.PyPlotBackend)
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:882
 [10] backend
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:245 [inlined]
 [11] pyplot()
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:86
 [12] top-level scope
    @ In[3]:3
In [4]:
## Let's use the standard iteration
using Plots
pyplot()

x = y = range(-25, stop = 25, length = 50)
x0 = [0.1,0.2]
p = plot(x,y,(x,y) -> quadratic(A,b,[x,y]), st=:surface,camera=(80,30))
plot!([x0[1]], [x0[2]], [quadratic(A,b,x0)])
α = 1.0
x = copy(x0)
for i=1:15
  g = A*x - b
  x .+= -α*g
  push!(p[1][2], x[1],x[2], quadratic(A,b,x))
  gui()
  sleep(0.25)
end

ArgumentError: Package PyPlot not found in current path.
- Run `import Pkg; Pkg.add("PyPlot")` to install the PyPlot package.

Stacktrace:
  [1] macro expansion
    @ ./loading.jl:1772 [inlined]
  [2] macro expansion
    @ ./lock.jl:267 [inlined]
  [3] __require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1753
  [4] #invoke_in_world#3
    @ ./essentials.jl:926 [inlined]
  [5] invoke_in_world
    @ ./essentials.jl:923 [inlined]
  [6] require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1746
  [7] top-level scope
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:883
  [8] eval
    @ ./boot.jl:385 [inlined]
  [9] _initialize_backend(pkg::Plots.PyPlotBackend)
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:882
 [10] backend
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:245 [inlined]
 [11] pyplot()
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:86
 [12] top-level scope
    @ In[4]:3
In [5]:
## Let's use a smaller value of alpha
using Plots
pyplot()

x = y = range(-1, stop = 1, length = 25)
x0 = randn(2)
p = plot(x,y,(x,y) -> quadratic(A,b,[x,y]), st=:surface,camera=(150,30), xlim=(-1,1), ylim=(-1,1), zlim=(0,5))
plot!([x0[1]], [x0[2]], [quadratic(A,b,x0)])
α = 0.1
x = copy(x0)
for i=1:15
  g = A*x - b
  x .+= -α*g
  push!(p[1][2], x[1],x[2], quadratic(A,b,x))
  gui()
  sleep(0.25)
end

ArgumentError: Package PyPlot not found in current path.
- Run `import Pkg; Pkg.add("PyPlot")` to install the PyPlot package.

Stacktrace:
  [1] macro expansion
    @ ./loading.jl:1772 [inlined]
  [2] macro expansion
    @ ./lock.jl:267 [inlined]
  [3] __require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1753
  [4] #invoke_in_world#3
    @ ./essentials.jl:926 [inlined]
  [5] invoke_in_world
    @ ./essentials.jl:923 [inlined]
  [6] require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1746
  [7] top-level scope
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:883
  [8] eval
    @ ./boot.jl:385 [inlined]
  [9] _initialize_backend(pkg::Plots.PyPlotBackend)
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:882
 [10] backend
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:245 [inlined]
 [11] pyplot()
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:86
 [12] top-level scope
    @ In[5]:3
In [6]:
## Let's use the optimal alpha
using Plots
pyplot()

x = y = range(-1, stop = 1, length = 25)
x0 = randn(2)
p = plot(x,y,(x,y) -> quadratic(A,b,[x,y]), st=:surface,camera=(150,30), xlim=(-1,1), ylim=(-1,1), zlim=(0,5))
plot!([x0[1]], [x0[2]], [quadratic(A,b,x0)])

x = copy(x0)
for i=1:15
  g = A*x - b
  α = g'*g / (g'*A*g)
  x .+= -α*g
  push!(p[1][2], x[1],x[2], quadratic(A,b,x))
  gui()
  sleep(0.25)
end

ArgumentError: Package PyPlot not found in current path.
- Run `import Pkg; Pkg.add("PyPlot")` to install the PyPlot package.

Stacktrace:
  [1] macro expansion
    @ ./loading.jl:1772 [inlined]
  [2] macro expansion
    @ ./lock.jl:267 [inlined]
  [3] __require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1753
  [4] #invoke_in_world#3
    @ ./essentials.jl:926 [inlined]
  [5] invoke_in_world
    @ ./essentials.jl:923 [inlined]
  [6] require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1746
  [7] top-level scope
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:883
  [8] eval
    @ ./boot.jl:385 [inlined]
  [9] _initialize_backend(pkg::Plots.PyPlotBackend)
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:882
 [10] backend
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:245 [inlined]
 [11] pyplot()
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:86
 [12] top-level scope
    @ In[6]:3
In [7]:
## Now, we can try coordinate descent
using Plots
pyplot()

x = y = range(-1, stop = 1, length = 25)
x0 = randn(2)
p = plot(x,y,(x,y) -> quadratic(A,b,[x,y]), st=:surface,camera=(150,30), xlim=(-1,1), ylim=(-1,1), zlim=(0,5))
plot!([x0[1]], [x0[2]], [quadratic(A,b,x0)])

x = copy(x0)
n = 2
for j=1:10
  i = mod(j,n)+1
  r = A*x - b
  x[i] += -r[i]/A[i,i]
  push!(p[1][2], x[1],x[2], quadratic(A,b,x))
  gui()
  sleep(0.25)
end

ArgumentError: Package PyPlot not found in current path.
- Run `import Pkg; Pkg.add("PyPlot")` to install the PyPlot package.

Stacktrace:
  [1] macro expansion
    @ ./loading.jl:1772 [inlined]
  [2] macro expansion
    @ ./lock.jl:267 [inlined]
  [3] __require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1753
  [4] #invoke_in_world#3
    @ ./essentials.jl:926 [inlined]
  [5] invoke_in_world
    @ ./essentials.jl:923 [inlined]
  [6] require(into::Module, mod::Symbol)
    @ Base ./loading.jl:1746
  [7] top-level scope
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:883
  [8] eval
    @ ./boot.jl:385 [inlined]
  [9] _initialize_backend(pkg::Plots.PyPlotBackend)
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:882
 [10] backend
    @ ~/.julia/packages/Plots/du2dt/src/backends.jl:245 [inlined]
 [11] pyplot()
    @ Plots ~/.julia/packages/Plots/du2dt/src/backends.jl:86
 [12] top-level scope
    @ In[7]:3
In [8]:
## A fun thing that Julia does