Moved solution from exercises/ to solutions/

exercises/Exercises 26.ipynb

 %% Cell type:markdown id: tags:
 
 # Exercises 26
 
 # problem 1
 
 Implement Newton's method for functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ given as a function of the form
 
 ```haskell
 newton :: (Vector Double -> Vector Double) -> (Vector Double -> Matrix Double) -> Vector Double -> Vector Double
 newton f df initial = [...]
 ```
 
 Here, `f` is the function whose root should be found, and `df` is its derivative (i.e. Jacobian). `initial` is an initial guess, and should return the first iterate for which the norm of the function value is smaller than some fixed bound.
 
-%% Cell type:code id: tags:
-
-``` haskell
-import Numeric.Container
-
-newton :: (Vector Double -> Vector Double) -> (Vector Double -> Matrix Double)
-       -> Vector Double -> Vector Double
-newton f df initial = fst . head . filter stoprule $ iterate step (initial, f initial)
-    where step (x, y) = let x' = x - df x <\> y
-                        in (x', f x')
-          stoprule :: (Vector Double, Vector Double) -> Bool
-          stoprule (_, y) = norm2 y < 1e-6
-```
-
-%% Cell type:code id: tags:
-
-``` haskell
-f :: Vector Double -> Vector Double
-f x = fromList [ x0^2 + x1^2
-               , x0^2 - x1^2]
-    where x0 = x @> 0
-          x1 = x @> 1
-
-df :: Vector Double -> Matrix Double
-df x = fromLists [ [ 2*x0,  2*x1 ]
-                 , [ 2*x0, -2*x1 ] ]
-     where x0 = x @> 0
-           x1 = x @> 1
-
-newton f df (fromList [10, 20])
-```
-
-%%%% Output: display_data
-
-
 %% Cell type:markdown id: tags:
 
 # problem 2
 
 (This is problem 4 of the SIAM 100-digit challenge.)
 
 Consider the function
 
 $$f(x, y) = e^{\sin(50x)} + \sin(60 e^y) + \sin(70\sin(x)) + \sin(\sin(80y)) - \sin(10(x+y)) + \frac{x^2 + y^2}{4}.$$
 
 1. Plot the function using the `vis.py` script from the source repository.
 
 2. Try to find the global minimum of the function by solving the equation
 
     $$\nabla f(x, y) = 0$$
 
     using Newton's method from problem 1.
 
     Since the function oscillates very quickly and therefore has multiple local extrema, you should solve the equation for a sufficiently large number of initial values. Use a grid of $10 \times 10$ points in the square $(-0.5,\, 0.5)^2 \subset \mathbb{R}^2$ and run Newton's method on each of these points.
 
 3. It may also be useful to try to refine the set of initial points until some heuristic criterion is fulfilled. Try to formulate such a criterion.
 
 For reference, the global minimum is -3.306868647.
 
 The function, its gradient and is Hessian are are implemented by the following Haskell functions:
 
 ```haskell
 f :: Double -> Double -> Double
 f x y = exp (sin (50*x))
       + sin (60 * exp y)
       + sin (70 * sin x)
       + sin (sin (80*y))
       - sin (10 * (x+y))
       + (x^2 + y^2)/4
 
 gradf :: Double -> Double -> [Double]
 gradf x y = [dfx, dfy]
     where dfx = 50 * cos (50*x) * exp (sin (50*x))
               + 70 * cos x * cos (70 * sin x)
               - 10 * cos (10 * (x+y))
               + x/2
           dfy = 60 * exp y * cos (60 * exp y)
               + 80 * cos (80*y) * cos (sin (80*y))
               - 10 * cos (10 * (x+y))
               + y/2
 
 hessf :: Double -> Double -> [[Double]]
 hessf x y = [[hfxx, hfxy], [hfxy, hfyy]]
     where hfxx = 2500 * cos (50*x) ^ 2 * exp (sin (50*x))
                - 2500 * sin (50*x) * exp (sin (50*x))
                - 70 * sin x * cos (70 * sin x)
                - 4900 * cos x ^ 2 * sin (70 * sin x)
                + 100 * sin (10 * (x+y))
                + 1/2
           hfyy = 60 * exp y * cos (60 * exp y)
                - 3600 * exp y ^ 2 * sin (60 * exp y)
                - 6400 * sin (80*y) * cos (sin (80*y))
                - 6400 * cos (80*y) ^ 2 * sin (sin (80*y))
                + 100 * sin(10 * (x+y))
                + 1/2
           hfxy = 100 * sin (10 * (x+y))
 ```

solutions/Exercises 26_sol.ipynb

 %% Cell type:markdown id: tags:
 
 # Exercises 26
 
 # problem 1
 
 Implement Newton's method for functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ given as a function of the form
 
 ```haskell
 newton :: (Vector Double -> Vector Double) -> (Vector Double -> Matrix Double) -> Vector Double -> Vector Double
 newton f df initial = [...]
 ```
 
 Here, `f` is the function whose root should be found, and `df` is its derivative (i.e. Jacobian). `initial` is an initial guess, and should return the first iterate for which the norm of the function value is smaller than some fixed bound.
 
+%% Cell type:code id: tags:
+
+``` haskell
+import Numeric.Container
+
+newton :: (Vector Double -> Vector Double) -> (Vector Double -> Matrix Double)
+       -> Vector Double -> Vector Double
+newton f df initial = fst . head . filter stoprule $ iterate step (initial, f initial)
+    where step (x, y) = let x' = x - df x <\> y
+                        in (x', f x')
+          stoprule :: (Vector Double, Vector Double) -> Bool
+          stoprule (_, y) = norm2 y < 1e-6
+```
+
+%% Cell type:code id: tags:
+
+``` haskell
+f :: Vector Double -> Vector Double
+f x = fromList [ x0^2 + x1^2
+               , x0^2 - x1^2]
+    where x0 = x @> 0
+          x1 = x @> 1
+
+df :: Vector Double -> Matrix Double
+df x = fromLists [ [ 2*x0,  2*x1 ]
+                 , [ 2*x0, -2*x1 ] ]
+     where x0 = x @> 0
+           x1 = x @> 1
+
+newton f df (fromList [10, 20])
+```
+
+%%%% Output: display_data
+
+
 %% Cell type:markdown id: tags:
 
 # problem 2
 
 (This is problem 4 of the SIAM 100-digit challenge.)
 
 Consider the function
 
 $$f(x, y) = e^{\sin(50x)} + \sin(60 e^y) + \sin(70\sin(x)) + \sin(\sin(80y)) - \sin(10(x+y)) + \frac{x^2 + y^2}{4}.$$
 
 1. Plot the function using the `vis.py` script from the source repository.
 
 2. Try to find the global minimum of the function by solving the equation
 
     $$\nabla f(x, y) = 0$$
 
     using Newton's method from problem 1.
 
     Since the function oscillates very quickly and therefore has multiple local extrema, you should solve the equation for a sufficiently large number of initial values. Use a grid of $10 \times 10$ points in the square $(-0.5,\, 0.5)^2 \subset \mathbb{R}^2$ and run Newton's method on each of these points.
 
 3. It may also be useful to try to refine the set of initial points until some heuristic criterion is fulfilled. Try to formulate such a criterion.
 
 For reference, the global minimum is -3.306868647.
 
 The function, its gradient and is Hessian are are implemented by the following Haskell functions:
 
 ```haskell
 f :: Double -> Double -> Double
 f x y = exp (sin (50*x))
       + sin (60 * exp y)
       + sin (70 * sin x)
       + sin (sin (80*y))
       - sin (10 * (x+y))
       + (x^2 + y^2)/4
 
 gradf :: Double -> Double -> [Double]
 gradf x y = [dfx, dfy]
     where dfx = 50 * cos (50*x) * exp (sin (50*x))
               + 70 * cos x * cos (70 * sin x)
               - 10 * cos (10 * (x+y))
               + x/2
           dfy = 60 * exp y * cos (60 * exp y)
               + 80 * cos (80*y) * cos (sin (80*y))
               - 10 * cos (10 * (x+y))
               + y/2
 
 hessf :: Double -> Double -> [[Double]]
 hessf x y = [[hfxx, hfxy], [hfxy, hfyy]]
     where hfxx = 2500 * cos (50*x) ^ 2 * exp (sin (50*x))
                - 2500 * sin (50*x) * exp (sin (50*x))
                - 70 * sin x * cos (70 * sin x)
                - 4900 * cos x ^ 2 * sin (70 * sin x)
                + 100 * sin (10 * (x+y))
                + 1/2
           hfyy = 60 * exp y * cos (60 * exp y)
                - 3600 * exp y ^ 2 * sin (60 * exp y)
                - 6400 * sin (80*y) * cos (sin (80*y))
                - 6400 * cos (80*y) ^ 2 * sin (sin (80*y))
                + 100 * sin(10 * (x+y))
                + 1/2
           hfxy = 100 * sin (10 * (x+y))
 ```