Commit ffd5e06d authored by Christoph Ruegge's avatar Christoph Ruegge
Browse files

move bfgs to archive

parent e1eaaddb
%% Cell type:markdown id: tags:
## minimization
## Minimization
%% Cell type:code id: tags:
``` haskell
import Numeric.GSL.Minimization
```
%% Cell type:markdown id: tags:
**minimize** <br /> Simplex algorithm (does not need a derivative).
```haskell
minimize
:: MinimizeMethod -- minimize method
-> Double -- desired precision of the solution (size test)
-> Int -- maximum number of iterations allowed
-> [Double] -- sizes of the initial search box
-> ([Double] -> Double) -- function to minimize
-> [Double] -- starting point
-> ([Double], Matrix Double) -- solution vector and optimization path
```
```haskell
data MinimizeMethod = NMSimplex
NMSimplex2
```
%% Cell type:code id: tags:
``` haskell
f [x,y] = 10*(x-1)^2 + 20*(y-2)^2 + 30
minimize NMSimplex2 1E-2 30 [1,1] f [5,7]
```
%%%% Output: display_data
%% Cell type:markdown id: tags:
Andere Verfahren wie Broyden-Fletcher-Goldfarb-Shanno (BFGS) in `minimizeD` (mit Ableitung)
## Finding roots
%% Cell type:code id: tags:
``` haskell
import Numeric.GSL.Root
```
%% Cell type:markdown id: tags:
**root** <br /> non-linear, multidimensional root finding (no derivative needed).
```haskell
root
:: RootMethod -- Methode
-> Double -- maximum residual
-> Int -- maximum number of iterations allowed
-> ([Double] -> [Double]) -- function to minimize
-> [Double] -- starting point
-> ([Double], Matrix Double) -- solution vector and optimization path
```
```haskell
data RootMethod = Hybrids
| Hybrid
| DNewton
| Broyden
```
%% Cell type:code id: tags:
``` haskell
(sol,path) = root Hybrid 1E-7 30 (\[x] -> [x^2 -1]) [-10]
print path
print sol
```
%%%% Output: display_data
%%%% Output: display_data
%% Cell type:markdown id: tags:
## Newton-method
%% Cell type:code id: tags:
``` haskell
newton :: (Fractional a, Floating a ) => (a -> a) -> a -> [(a,a)]
newton f xn = (f xn,xn) : newton f (fn xn)
where fn xn = let h = 1e-11 in xn - f xn/((f (xn+h) - f xn)/h)
```
%% Cell type:code id: tags:
``` haskell
unzip $ takeWhile (\(y,_) -> abs y >1e-16) $ newton cos (1::Double)
import Numeric.Container
import Graphics.Rendering.Chart.Easy hiding (Matrix, Vector, scale)
import Graphics.Rendering.Chart.Backend.Cairo
(f,_) = unzip $ take 10 $ newton cos (1::Double)
toRenderable $ do
layout_title .= "Solution"
setColors [opaque blue, opaque red]
plot ( line "f" [ zip [0..] (map LogValue f) ] )
plot ( points "points" ( zip [0..] (map LogValue f) ) )
```
%%%% Output: display_data
%%%% Output: display_data
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%% Cell type:code id: tags:
``` haskell
import Numeric.Container
import Numeric.LinearAlgebra.Util
import Data.Complex
n = 100 -- size of the domain
-- create matrix of the mesh (and newton iteration start values)
coord = linspace n (-0.9,0.9::Double)
m = buildMatrix n n (\(i,j) -> (coord @> i) :+ coord @> j) :: Matrix (Complex Double)
```
%% Cell type:code id: tags:
``` haskell
import Data.List (genericLength)
f x = x**3 - 1
nm = snd . unzip . takeWhile (\(y,_) -> magnitude y >1e-13) . newton f
-- solution of newton iteration for every startvalue
solm = mapMatrix (last . nm ) m
phasem = mapMatrix phase solm
-- number of iterations for every startvalue
numitm = mapMatrix (genericLength . nm ) m :: Matrix Double
```
%% Cell type:code id: tags:
``` haskell
import Graphics.Rendering.Plot
figsol = do
withTitle $ setText "Newton fractal for x^3 - 1"
withSubTitle $ setText "phase of solution"
setPlots 1 1
withPlot (1,1) $ do
setDataset phasem
addAxis XAxis (Side Lower) $ setTickLabelFormat DefaultTickFormat
addAxis YAxis (Side Lower) $ setTickLabelFormat DefaultTickFormat
setRangeFromData XAxis Lower Linear
setRangeFromData YAxis Lower Linear
figit = do
withTitle $ setText "Newton fractal for x^3 - 1"
withSubTitle $ setText "number of iterations"
setPlots 1 1
withPlot (1,1) $ do
setDataset numitm
addAxis XAxis (Side Lower) $ setTickLabelFormat DefaultTickFormat
addAxis YAxis (Side Lower) $ setTickLabelFormat DefaultTickFormat
setRangeFromData XAxis Lower Linear
setRangeFromData YAxis Lower Linear
figsol
figit
```
%%%% Output: display_data
%% Cell type:code id: tags:
``` haskell
saveMatrix "xmat.txt" "%f" $ mapMatrix realPart m
saveMatrix "ymat.txt" "%f" $ mapMatrix imagPart m
saveMatrix "zmat.txt" "%f" numitm
-- ../vis.py -x xmat.txt -y ymat.txt -z zmat.txt -t image
```
%%%% Output: display_data
%% Cell type:markdown id: tags:
<img src="../images/newton_konvergenz.png"></img>
<img src="../images/newton_conv_sol.png"></img>
%% Cell type:markdown id: tags:
Solve the following non-linear least squares problem:
$$(P) \qquad \min_{\mathbb{R}^7}{f(x)\equiv \frac{1}{2}\|g(x)\|_2^2}$$
with $g: \mathbb{R}^7 \rightarrow \mathbb{R}^8$, $\nabla g : \mathbb{R}^7 \rightarrow\mathbb{R}^{7\times 8}$ and
$\nabla^2 g: \mathbb{R}^7 \rightarrow \mathbb{R}^{7\times 7\times 8}$ .
Given initial point $x_0=zeros(7,1)$ find a approximation $\tilde{x}$ for a global solution with $\|\nabla f(\tilde{x})\|\leq 10^{-11}$.
- plot of norm of the gradient ,
- plot of function values,
- plot of stepsize,
<!--
plot of Funktionswert, Gradient und Hessian (Tabelle) mit folgenden Hessians:
- $H_{\nu}=\nabla^2 f(x_\nu)$
-->
in every step you have to solve a linear system
- Newton-method
- BFGS
%% Cell type:code id: tags:
``` haskell
import Numeric.Container
g :: Vector Double -> Vector Double
g x = fromList [ 105-4*x @> 0 -5*x @> 1+3*x@> 6**2,
-10*x@>0+8*x@>2+x@>6,
8*x@>0-2*x@>1+12*x@>4-5,
-3*(x@>0-2)**2 -4*(x@>1-3)**2-2*x@>2**2+7*x@>3+100,
-5*x@>0**2-8*x@>1-(x@>2-6)**2+2*x@>3+40,
-0.5*(x@>0-8)**2-2*(x@>1-4)**2-3*x@>4**2+x@>5+30,
-x@>0**2-2*(x@>1-2)**2+2*x@>0*x@>1-14*x@>4+6*x@>5,
3*x@>0-6*x@>1-12*x@>4*x@>5 ]
f :: Vector Double -> Double
func x = 0.5 * (norm $ g x)**2
--"""gradfunc is the gradient of func(x)
-- norm(g(x))*norm(g(x)) *gradg """
gradfunc x = norm $ gradg x
--"""gradg is the gradient of g(x).column vectors as arrays"""
gradg x = multiply (trans $ Jg x) g x
--""" hessf is the Hessian of the function func(x)=(1/2)*norm(g(x))^2 """
hessf x =
[h1,h2,h3,h4,h5,h6,h7,h8] = Hg(x)
z = g x
multiply (trans $ Jg x) Jg x + z@>0*h1+z@>1*h2+z@>2*h3+z@>3*h4+z@>4*h5+z@>5*h6+z@>6*h7+z@>7*h8
-- """ this function file gives the Hessians of the component functions of g """
Hg x =
h1 = np.diag((0,0,0,0,0,0,6))
h2 = np.zeros((7,7))
h3 = h2
h4 = np.diag((-6,-8,-4,0,0,0,0))
h5 = np.diag((-10,0,-2,0,0,0,0))
h6 = np.diag((-1,-4,0,0,-6,0,0))
h7 = np.diag((-2,-4,0,0,0,0,0))+ np.diag((2,0,0,0,0,0),k=1)+ np.diag((2,0,0,0,0,0),k=-1)
h8 = np.diag((0,0,0,0,-12,0),k=1) + np.diag((0,0,0,0,-12,0),k=-1)
return (h1,h2,h3,h4,h5,h6,h7,h8)
-- """ this one does the Jacobian of the sample function g. """
Jg x = fromLists [[-4,-5 ,0, 0, 0, 0, 6*x@>6],
[-10, 0, 8, 0, 0, 0, 1],
[8, -2, 0, 0, 12, 0, 0],
[-6*(x@>0-2), -8*(x@>1-3), -4*x@>2, 7, 0, 0, 0],
[-10*x@>0, -8, -2*(x@>2-6), 2, 0, 0, 0],
[-(x@>0-8), -4*(x@>1-4), 0, 0, -6*x@>4, 1, 0],
[-2*x@>0+2*x@>1, -4*(x@>1-2)+2*x@>0, 0, 0, -14, 6, 0],
[3, -6, 0, 0, -12*x@>5, -12*x@>4, 0]]
```
%%%% Output: display_data
%%%% Output: display_data
%% Cell type:code id: tags:
``` haskell
g $ fromList [1,1,2,3,4,5,6]
```
%%%% Output: display_data
%% Cell type:code id: tags:
``` haskell
```
......
%% Cell type:markdown id: tags:
Solve the following non-linear least squares problem:
$$(P) \qquad \min_{\mathbb{R}^7}{f(x)\equiv \frac{1}{2}\|g(x)\|_2^2}$$
with $g: \mathbb{R}^7 \rightarrow \mathbb{R}^8$, $\nabla g : \mathbb{R}^7 \rightarrow\mathbb{R}^{7\times 8}$ and
$\nabla^2 g: \mathbb{R}^7 \rightarrow \mathbb{R}^{7\times 7\times 8}$ .
Given initial point $x_0=zeros(7,1)$ find a approximation $\tilde{x}$ for a global solution with $\|\nabla f(\tilde{x})\|\leq 10^{-11}$.
- plot of norm of the gradient ,
- plot of function values,
- plot of stepsize,
<!--
plot of Funktionswert, Gradient und Hessian (Tabelle) mit folgenden Hessians:
- $H_{\nu}=\nabla^2 f(x_\nu)$
-->
in every step you have to solve a linear system
- Newton-method
- BFGS
%% Cell type:code id: tags:
``` haskell
import Numeric.Container
g :: Vector Double -> Vector Double
g x = fromList [ 105-4*x @> 0 -5*x @> 1+3*x@> 6**2,
-10*x@>0+8*x@>2+x@>6,
8*x@>0-2*x@>1+12*x@>4-5,
-3*(x@>0-2)**2 -4*(x@>1-3)**2-2*x@>2**2+7*x@>3+100,
-5*x@>0**2-8*x@>1-(x@>2-6)**2+2*x@>3+40,
-0.5*(x@>0-8)**2-2*(x@>1-4)**2-3*x@>4**2+x@>5+30,
-x@>0**2-2*(x@>1-2)**2+2*x@>0*x@>1-14*x@>4+6*x@>5,
3*x@>0-6*x@>1-12*x@>4*x@>5 ]
f :: Vector Double -> Double
func x = 0.5 * (norm $ g x)**2
--"""gradfunc is the gradient of func(x)
-- norm(g(x))*norm(g(x)) *gradg """
gradfunc x = norm $ gradg x
--"""gradg is the gradient of g(x).column vectors as arrays"""
gradg x = multiply (trans $ Jg x) g x
--""" hessf is the Hessian of the function func(x)=(1/2)*norm(g(x))^2 """
hessf x =
[h1,h2,h3,h4,h5,h6,h7,h8] = Hg(x)
z = g x
multiply (trans $ Jg x) Jg x + z@>0*h1+z@>1*h2+z@>2*h3+z@>3*h4+z@>4*h5+z@>5*h6+z@>6*h7+z@>7*h8
-- """ this function file gives the Hessians of the component functions of g """
Hg x =
h1 = np.diag((0,0,0,0,0,0,6))
h2 = np.zeros((7,7))
h3 = h2
h4 = np.diag((-6,-8,-4,0,0,0,0))
h5 = np.diag((-10,0,-2,0,0,0,0))
h6 = np.diag((-1,-4,0,0,-6,0,0))
h7 = np.diag((-2,-4,0,0,0,0,0))+ np.diag((2,0,0,0,0,0),k=1)+ np.diag((2,0,0,0,0,0),k=-1)
h8 = np.diag((0,0,0,0,-12,0),k=1) + np.diag((0,0,0,0,-12,0),k=-1)
return (h1,h2,h3,h4,h5,h6,h7,h8)
-- """ this one does the Jacobian of the sample function g. """
Jg x = fromLists [[-4,-5 ,0, 0, 0, 0, 6*x@>6],
[-10, 0, 8, 0, 0, 0, 1],
[8, -2, 0, 0, 12, 0, 0],
[-6*(x@>0-2), -8*(x@>1-3), -4*x@>2, 7, 0, 0, 0],
[-10*x@>0, -8, -2*(x@>2-6), 2, 0, 0, 0],
[-(x@>0-8), -4*(x@>1-4), 0, 0, -6*x@>4, 1, 0],
[-2*x@>0+2*x@>1, -4*(x@>1-2)+2*x@>0, 0, 0, -14, 6, 0],
[3, -6, 0, 0, -12*x@>5, -12*x@>4, 0]]
```
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