From 97d03cdab3a080836fb32b3a8585f99a25740f49 Mon Sep 17 00:00:00 2001 From: rohan Date: Tue, 21 Nov 2017 02:06:41 +0530 Subject: [PATCH] updated romberg graphs --- week3_RombergInterpolation/comparison.png | Bin 53959 -> 35316 bytes .../comparison_splint.png | Bin 53663 -> 38215 bytes week3_RombergInterpolation/errorVScalls.png | Bin 42142 -> 33869 bytes week3_RombergInterpolation/loglog_fnplot.png | Bin 0 -> 30195 bytes .../qromb_errvscalls.png | Bin 35410 -> 30441 bytes .../qromb_errvscalls_split.png | Bin 38534 -> 31447 bytes week3_RombergInterpolation/romberg.py | 168 ++ week3_RombergInterpolation/rombergCalls.png | Bin 33179 -> 31575 bytes week3_RombergInterpolation/splineErr.png | Bin 43213 -> 36428 bytes .../splineErr_withSplit.png | Bin 49471 -> 36853 bytes .../week3_1-lyxformat-413.lyx~ | 1737 +++++++++++++++++ .../week3_1.html.LyXconv/comparison.png | Bin 54130 -> 35338 bytes .../comparison_splint.png | Bin 53736 -> 38154 bytes .../week3_1.html.LyXconv/errorVScalls.png | Bin 42280 -> 33856 bytes .../week3_1.html.LyXconv/fnplot.png | Bin 31441 -> 31428 bytes .../week3_1.html.LyXconv/fnplot_2.png | Bin 39365 -> 39352 bytes .../week3_1.html.LyXconv/loglog_fnplot.png | Bin 0 -> 30273 bytes .../week3_1.html.LyXconv/qromb_errvscalls.png | Bin 35438 -> 30430 bytes .../qromb_errvscalls_split.png | Bin 38683 -> 31459 bytes .../week3_1.html.LyXconv/rombergCalls.png | Bin 33248 -> 31548 bytes .../week3_1.html.LyXconv/splineErr.png | Bin 43236 -> 36296 bytes .../splineErr_withSplit.png | Bin 49545 -> 36784 bytes .../week3_1.html.LyXconv/week3_1.html | 29 +- week3_RombergInterpolation/week3_1.lyx | 48 +- week7_ProbabilityDistbn/surface3d_demo.py | 2 +- week8_Minimization/min2_1.png | Bin 0 -> 231852 bytes week8_Minimization/min2_cnt.png | Bin 0 -> 27181 bytes week8_Minimization/min2_cnt2.png | Bin 0 -> 33400 bytes week8_Minimization/min2asg.lyx | 1343 +++++++++++++ week8_Minimization/min2asg.lyx~ | 969 +++++++++ week8_Minimization/min2asg.py | 168 ++ week8_Minimization/spiral1.png | Bin 0 -> 8470 bytes week8_Minimization/spiral2.png | Bin 0 -> 14865 bytes week8_Minimization/spiral3.png | Bin 0 -> 17722 bytes week8_Minimization/spiral4.png | Bin 0 -> 37792 bytes 35 files changed, 4438 insertions(+), 26 deletions(-) create mode 100644 week3_RombergInterpolation/loglog_fnplot.png create mode 100644 week3_RombergInterpolation/romberg.py create mode 100644 week3_RombergInterpolation/week3_1-lyxformat-413.lyx~ create mode 100644 week3_RombergInterpolation/week3_1.html.LyXconv/loglog_fnplot.png create mode 100644 week8_Minimization/min2_1.png create mode 100644 week8_Minimization/min2_cnt.png create mode 100644 week8_Minimization/min2_cnt2.png create mode 100644 week8_Minimization/min2asg.lyx create mode 100644 week8_Minimization/min2asg.lyx~ create mode 100644 week8_Minimization/min2asg.py create mode 100644 week8_Minimization/spiral1.png create mode 100644 week8_Minimization/spiral2.png create mode 100644 week8_Minimization/spiral3.png 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zXVv4=dl;xlBf9eKJy7qn$#MI&2U_A*2d_p4;lTu(bMx|2gKw>s??fg_aClt7^vF1uLEbabcot-w3br9P(p)lK4KvQSNJxDAGNOLS zXydwd*T}z-@|o4dT!la-_~oQK8%acJAXn6HGe$}d!70?e?;lwg2mq*`HyR?EB0m(k lfqZo8&#Bq}-$9S(mkfzMe5^hisYZn#MLAX3Co=li{trMN5vBkD diff --git a/week3_RombergInterpolation/romberg.py b/week3_RombergInterpolation/romberg.py new file mode 100644 index 0000000..d4fe010 --- /dev/null +++ b/week3_RombergInterpolation/romberg.py @@ -0,0 +1,168 @@ +# -*- coding: utf-8 -*- +from pylab import * +import scipy.special as sp +from scipy import integrate +import romberg as r + +def func(u): + return 2*u*sp.jv(3,2.7*u)**2 + +def func2(u): + return 2*sp.kv(3,1.2*u)**2*u*abs(sp.jv(3,2.7)/sp.kv(3,1.2))**2 + +count=0 +def integrand(u): + global count + count+=1 + if u<1.0: + return func(u) + else: + return func2(u) + +intg1=np.vectorize(integrand) +x=logspace(-3,7,200) +#semilogx(x,integrand(x)) +loglog(x,intg1(x)) +title("Dielectric Fibre Electromagnetic Mode integrand function") +xlabel("Value of x ->") +ylabel("Value of function f(x) ->") +grid() +show() + +def exact(): + return sp.jv(3,2.7)**2-sp.jv(4,2.7)*sp.jv(2,2.7)+abs(sp.jv(3,2.7)/sp.kv(3,1.2))**2*(sp.kv(4,1.2)*sp.kv(2,1.2)-sp.kv(3,1.2)**2) + +I0=exact() +print I0 +print integrate.quad(integrand,0,20,full_output=0)[0]-exact() +#n_eval = 567 for non-split quad +#print integrate.quad(integrand,0,1,full_output=1), integrate.quad(integrand,1,20,full_output=1) +#n_eval = 147+21 = 168 for split quad +counts=[] +errs=[] +s=0 +""" +for i in range(1,20): + count=0 + s=r.trapzd(integrand,0,20,s,i) + print "%1d %.15f %.2e"%(i,s,s-I0) + print count + counts.append(count) + errs.append(abs(s-I0)) + +loglog(counts,errs) +loglog(counts,errs,'ro') +title("Error in integration vs number of function calls") +ylabel("Error in integration (absolute)") +xlabel("Number of function calls") +grid() +show() + +#print r.qromb(integrand,0,20,1e-10) +counts=[] +errs=[] +for i in range(-1,-11,-1): + count=0 + val= r.qromb(integrand,0,20,10**i) + print val + counts.append(count) + errs.append(abs(val[1])) + +loglog(counts,errs) +loglog(counts,errs,'ro') +title("Error vs number of function calls") +ylabel("Error in integration") +xlabel("Number of function calls") +grid() +#show() + + +counts=[] +errors=[] + +for i in range(-1,-12,-1): + count=0 + x1=r.qromb(integrand,0,1,10**i) + x2=r.qromb(integrand,1,20,10**i) + print x1[0]+x2[0],x1[0]+x2[0]-exact(),x1[2]+x2[2] + counts.append(count) + errors.append(abs(x1[0]+x2[0]-exact())) + +loglog(counts, errors) +loglog(counts,errors,'ro') +title("Error vs number of function calls (split 0-1 and 1-20)") +xlabel("Number of function calls") +ylabel("Error in integration") +#grid() +show() + +counts=[] +for i in range(5,21): + count=0 + x=r.qromb(integrand,0,20,1e-8,k=i) + print x + counts.append(count) + +semilogy(range(5,21),counts) +semilogy(range(5,21),counts,'ro') +title("Number of function calls vs order") +xlabel("Order of Romberg Integration") +ylabel("Number of function calls") +grid() +show() + +import scipy.interpolate as si +err=[] +for i in range(4,20): + x=linspace(0,20,2**i) + y=intg1(x) + tck=si.splrep(x,y) + I=si.splint(0,20,tck) + print I-exact() + err.append(abs(I-exact())) +semilogy(range(4,20),err) +semilogy(range(4,20),err,'ro') +err=[] +for i in range(4,20): + x1=linspace(0,1,2**(i-1)) + y1=intg1(x1) + tck1=si.splrep(x1,y1) + I1=si.splint(0,1,tck1) + x2=linspace(1,20,2**(i-1)) + y2=intg1(x2) + tck2=si.splrep(x2,y2) + I2=si.splint(1,20,tck2) + print I1+I2-exact() + err.append(abs(I1+I2-exact())) +semilogy(range(4,20),err) +semilogy(range(4,20),err,'ro') +title("Comparison of both methods") +xlabel("log2(number of points)") +ylabel("Error relative to actual integral value") +grid() +show() +""" + +def trap3(func, a, b, n): + if(n==1): + return 0.5*(b-a)*(func(a)+func(b)) + else: + d = (float)(b-a)/3**(n-1) + sum=0.0 + x=a+d + while(x+_%FYN8GK%aiE3(IZ zT>5MsQ;c2tPcsaapcgYvk?;U2NDsnZ=TB_6!4=Sj>iVSVq&bY)KDnY?c2r(C48o*Kj#l0Q7TXWe{eX}MKp2x*y 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For more info see http://www.lyx.org/ +\lyxformat 413 +\begin_document +\begin_header +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language english +\language_package default +\inputencoding auto +\fontencoding global +\font_roman default +\font_sans default +\font_typewriter default +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 +\font_tt_scale 100 + +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\use_hyperref false +\papersize default +\use_geometry false +\use_amsmath 1 +\use_esint 1 +\use_mhchem 1 +\use_mathdots 1 +\cite_engine basic +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\use_refstyle 1 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\quotes_language english +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +EE5011: Computer Methods in Electrical Engineering +\end_layout + +\begin_layout Title +Romberg Integration (Week 4) +\end_layout + +\begin_layout Author +Rohan Rao, EE14B118 +\end_layout + +\begin_layout Section +Programming Assignment +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +I=\frac{2}{a^{2}}\intop_{0}^{a}J_{v}^{2}(\mathrm{\kappa r)rdr+\frac{2}{a^{2}}\left|\frac{J_{v}(\kappa a)}{K_{v}(\text{γa)}}\right|^{2}\int_{a}^{\text{∞}}K_{v}^{2}(\gamma r)rdr} +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +Transformation using the dimensionless variable +\begin_inset Formula $u=\frac{r}{a}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $u=\frac{r}{a}$ +\end_inset + + means +\begin_inset Formula $r=a.u$ +\end_inset + + and +\begin_inset Formula $dr=a.du$ +\end_inset + +, r=0 means u=0 and r=a means u=1 +\end_layout + +\begin_layout Standard +On substituting, +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +I=2\intop_{0}^{1}J_{v}^{2}(\mathrm{ku)udu+\frac{2}{a^{2}}\left|\frac{J_{v}(k)}{K_{v}(\text{g)}}\right|^{2}\int_{1}^{\text{∞}}K_{v}^{2}(gu)udu} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Where k= +\begin_inset Formula $\kappa a=2.7$ +\end_inset + + and +\begin_inset Formula $g=\gamma a=1.2$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Plot the integrand in Python +\end_layout + +\begin_layout Standard +The integrand is composed of two parts: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +f(u)=2*J_{v}^{2}(ku)*u +\] + +\end_inset + + for +\begin_inset Formula $0\leq u<1$ +\end_inset + + and +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +f(u)=2*\left|\frac{J_{v}^{2}(k)}{K_{v}^{2}(g)}\right|*K_{v}^{2}(gu)*u +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +for +\begin_inset Formula $1\leq u<\text{∞}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +The plot of the function on a semi log scale (in u) is as follows: +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename fnplot.png + +\end_inset + + +\end_layout + +\begin_layout Standard +On a semi log scale in y = f(u) on the range [0.001,10] is as follows: +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename fnplot_2.png + +\end_inset + + +\end_layout + +\begin_layout Standard +The above plot shows that beyond x=2 (approx), the function behaves like + an exponentially decaying function, and by the time x=10, the value has + dropped by 10 orders of magnitude. +\end_layout + +\begin_layout Standard +Code: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +def func(u): +\end_layout + +\begin_layout Plain Layout + + return 2*u*sp.jv(3,2.7*u)**2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def func2(u): +\end_layout + +\begin_layout Plain Layout + + return 2*sp.kv(3,1.2*u)**2*u*abs(sp.jv(3,2.7)/sp.kv(3,1.2))**2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def integrand(u): +\end_layout + +\begin_layout Plain Layout + + if u<1.0: +\end_layout + +\begin_layout Plain Layout + + return func(u) +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + return func2(u) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +x=logspace(-3,7,200) +\end_layout + +\begin_layout Plain Layout + +semilogx(x,integrand(x)) +\end_layout + +\begin_layout Plain Layout + +title("Dielectric Fibre Electromagnetic Mode integrand function") +\end_layout + +\begin_layout Plain Layout + +xlabel("Value of x ->") +\end_layout + +\begin_layout Plain Layout + +ylabel("Value of function f(x) ->") +\end_layout + +\begin_layout Plain Layout + +show() +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The function reaches a maximum value of nearly 0.13 at the point x=1. + This is not visible even in the plot above, since the logspace() function + defined above does not include the sample x=1. + This shows that even finer sampling will be required in that region. + The function is continuous, as can be verified from checking the value + from the above func() and func2() at x=1. + The function is not smooth at x=1 since it has a very sharp variation in + the second derivative, forming a discontinuity. + This will require a very small value of h when using trapzd integration. +\end_layout + +\begin_layout Subsection +Analytically evaluating the integral +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int_{a}^{\text{∞}}f(u)du +\] + +\end_inset + + The function func2() is the value of the integrand beyond u=1. + Writing it in the convenient form +\begin_inset Formula $f(u)=C*K_{v}(1.2u)*u$ +\end_inset + +, where +\begin_inset Formula $C=2*\left|\frac{J_{v}^{2}(k)}{K_{v}^{2}(k)}\right|,v=3,k=2.7,g=1.2$ +\end_inset + +: +\end_layout + +\begin_layout Standard +For +\begin_inset Formula $x\ggg v$ +\end_inset + +, +\begin_inset Formula $K_{v}(x)$ +\end_inset + + can be approximated asymptotically by the function +\begin_inset Formula $\frac{\pi}{\sqrt{2\pi x}}e^{-x}$ +\end_inset + +. + Thus: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int_{a}^{\text{∞}}f(u)du=\int_{a}^{\text{∞}}\left(C*u*\frac{\pi}{\sqrt{2\pi*1.2u}}\exp(-1.2u)\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +This simplifies to: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int_{a}^{\text{∞}}C_{2}\sqrt{u}\exp(-1.2u)du +\] + +\end_inset + + where +\begin_inset Formula $C_{2}=C\sqrt{\frac{\pi}{2.4}}$ +\end_inset + +. + For large values of u, we can upper-bound the above integral by taking + +\begin_inset Formula $\sqrt{u}<\sqrt{100}<10.$ +\end_inset + + Thus, from the above functions, since we can ascertain that the function + value will drop to below +\begin_inset Formula $10^{-107}$ +\end_inset + + by u=100, we can bound the integral by using the above approximation. + The final form of the expression is as follows: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\int_{a}^{\text{∞}}f(u)du<\int_{a}^{\text{∞}}10.C_{2}\exp(-1.2u)du=\frac{10C_{2}}{1.2}\exp(-1.2a) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Substituting for C=0.00844 and +\begin_inset Formula $C_{2}=C\sqrt{\frac{\pi}{2.4}}$ +\end_inset + + we get the value of the above expression as 0.08046*exp(-1.2a). + We can evaluate this for various values of +\begin_inset Quotes eld +\end_inset + +a +\begin_inset Quotes erd +\end_inset + + so as to obtain the point beyond which the above integration is below the + required tolerance of the problem. + For the purpose of the following set of questions, evaluating the function + at a=15 and a=20 gives values of the order of +\begin_inset Formula $10^{-9}$ +\end_inset + + and +\begin_inset Formula $10^{-12}$ +\end_inset + + respectively. + So, a=20 is sufficient for a required tolerance of +\begin_inset Formula $10^{-10}$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Using scipy.integrate.quad to do the integration +\end_layout + +\begin_layout Standard +First of all, the exact value of the integral (derived analytically, provided + beforehand) is as follows: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +def exact(): +\end_layout + +\begin_layout Plain Layout + +return sp.jv(3,2.7)**2-sp.jv(4,2.7)*sp.jv(2,2.7)+abs(sp.jv(3,2.7)/sp.kv(3,1.2))**2*(sp.kv(4 +,1.2)*sp.kv(2,1.2)-sp.kv(3,1.2)**2) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +If we use scipy.integrate on the integrand function f(u) by considering x=20 + as the upper limit of the integral described above, we get: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +def func(u): +\end_layout + +\begin_layout Plain Layout + + return 2*u*sp.jv(3,2.7*u)**2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def func2(u): +\end_layout + +\begin_layout Plain Layout + + return 2*sp.kv(3,1.2*u)**2*u*abs(sp.jv(3,2.7)/sp.kv(3,1.2))**2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def integrand(u): +\end_layout + +\begin_layout Plain Layout + + if u<1.0: +\end_layout + +\begin_layout Plain Layout + + return func(u) +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + return func2(u) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +In: integrate.quad(integrand,0,20) +\end_layout + +\begin_layout Plain Layout + +Out: (0.04603886028000008, 6.923921130885802e-09) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Upon turning on full_output=1, number of function evaluations required shows + as neval=567 for the upper limit of x=20. + The error between the obtained value and the exact analytic expression + can be obtained by subtracting exact(). +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +In: integrate.quad(integrand,0,20,full_output=0)[0]-exact() +\end_layout + +\begin_layout Plain Layout + +Out:1.3555823130673161e-13 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +This shows that the upper limit of a=20 is sufficient for obtaining an accurate + answer. +\end_layout + +\begin_layout Subsection +Using trapzd from the Romberg module +\end_layout + +\begin_layout Standard +Using a global variable count and then calculating the number of function + calls in a trapzd loop, we get the following outputs: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +for i in range(1,20): +\end_layout + +\begin_layout Plain Layout + + s=r.trapzd(integrand,0,20,s,i) +\end_layout + +\begin_layout Plain Layout + + print "%1d %.15f %.2e"%(i,s,s-I0) +\end_layout + +\begin_layout Plain Layout + + print count +\end_layout + +\begin_layout Plain Layout + + count=0 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +Out: +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + 2, +\end_layout + +\begin_layout Plain Layout + + 1, +\end_layout + +\begin_layout Plain Layout + + 2, +\end_layout + +\begin_layout Plain Layout + + 4, +\end_layout + +\begin_layout Plain Layout + + 8, +\end_layout + +\begin_layout Plain Layout + + 16, +\end_layout + +\begin_layout Plain Layout + + 32, +\end_layout + +\begin_layout Plain Layout + + 64, +\end_layout + +\begin_layout Plain Layout + + 128, +\end_layout + +\begin_layout Plain Layout + + 256, +\end_layout + +\begin_layout Plain Layout + + 512, +\end_layout + +\begin_layout Plain Layout + + 1024, +\end_layout + +\begin_layout Plain Layout + + 2048, +\end_layout + +\begin_layout Plain Layout + + 4096, +\end_layout + +\begin_layout Plain Layout + + 8192, +\end_layout + +\begin_layout Plain Layout + + 16384, +\end_layout + +\begin_layout Plain Layout + + 32768, +\end_layout + +\begin_layout Plain Layout + + 65536, +\end_layout + +\begin_layout Plain Layout + + 131072 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +Out2: +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +1 0.000000000000000 -4.60e-02 +\end_layout + +\begin_layout Plain Layout + +2 0.000000000008383 -4.60e-02 +\end_layout + +\begin_layout Plain Layout + +3 0.000001289283323 -4.60e-02 +\end_layout + +\begin_layout Plain Layout + +4 0.000787977244362 -4.53e-02 +\end_layout + +\begin_layout Plain Layout + +5 0.044751328048882 -1.29e-03 +\end_layout + +\begin_layout Plain Layout + +6 0.029521687183219 -1.65e-02 +\end_layout + +\begin_layout Plain Layout + +7 0.046897366417297 8.59e-04 +\end_layout + +\begin_layout Plain Layout + +8 0.044868891255571 -1.17e-03 +\end_layout + +\begin_layout Plain Layout + +9 0.046056219282115 1.74e-05 +\end_layout + +\begin_layout Plain Layout + +10 0.045961594571426 -7.73e-05 +\end_layout + +\begin_layout Plain Layout + +11 0.046040763026165 1.90e-06 +\end_layout + +\begin_layout Plain Layout + +12 0.046034076754719 -4.78e-06 +\end_layout + +\begin_layout Plain Layout + +13 0.046038966706764 1.06e-07 +\end_layout + +\begin_layout Plain Layout + +14 0.046038560504368 -3.00e-07 +\end_layout + +\begin_layout Plain Layout + +15 0.046038867128737 6.85e-09 +\end_layout + +\begin_layout Plain Layout + +16 0.046038841556128 -1.87e-08 +\end_layout + +\begin_layout Plain Layout + +17 0.046038860704844 4.25e-10 +\end_layout + +\begin_layout Plain Layout + +18 0.046038859109436 -1.17e-09 +\end_layout + +\begin_layout Plain Layout + +19 0.046038860306469 2.66e-11 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The plot obtained (log-log plot due to large order variations): +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename errorVScalls.png + +\end_inset + + +\end_layout + +\begin_layout Standard +The overall trend shows that the function value is going down with increasing + number of function calls. + However, trapzd clearly requires 4-5 +\series bold +orders of magnitude +\series default + +\series bold +more +\series default +function calls than scipy.integrate.quad to obtain the same accuracy of nearly + +\begin_inset Formula $10^{-10}$ +\end_inset + +, and with 512 function calls, its error is nearly +\begin_inset Formula $2*10^{-6}$ +\end_inset + +. + It makes a difference whether r=1 is included in the function evaluation, + since the function value around that point drops off extremely quickly + and hence the integral value will differ based on it. +\end_layout + +\begin_layout Subsection +Using qromb from the Romberg module +\end_layout + +\begin_layout Standard +The use of qromb over the interval (0, a=20) with a required tolerance of + +\begin_inset Formula $10^{-10}$ +\end_inset + + gives the number of function calls as N=524289, 3 orders of magnitude greater + than scipy.quad.integrate. +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +In: r.qromb(integrand,0,20,1e-10) +\end_layout + +\begin_layout Plain Layout + +Out: (0.046038860123720424, -3.0025314137784918e-12, 524289) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The output of the following code will show how the error scales with the + number of function calls: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +In: for i in range(-1,-11,-1): +\end_layout + +\begin_layout Plain Layout + + print r.qromb(integrand,0,20,10**i) +\end_layout + +\begin_layout Plain Layout + +Out: +\end_layout + +\begin_layout Plain Layout + +(0.06453069939527602, 0.00024762829418648665, 17) +\end_layout + +\begin_layout Plain Layout + +(0.06453069939527602, 0.00024762829418648665, 17) +\end_layout + +\begin_layout Plain Layout + +(0.04666239432307808, 1.2538441623417591e-05, 257) +\end_layout + +\begin_layout Plain Layout + +(0.04588102573376104, -3.003242764428245e-06, 513) +\end_layout + +\begin_layout Plain Layout + +(0.04602855383209874, -1.9770401457648317e-07, 2049) +\end_layout + +\begin_layout Plain Layout + +(0.04603822202670924, -1.2278888980499384e-08, 8193) +\end_layout + +\begin_layout Plain Layout + +(0.04603902020149272, 3.069905837882252e-09, 16385) +\end_layout + +\begin_layout Plain Layout + +(0.04603887027120843, 1.9222654805821073e-10, 65537) +\end_layout + +\begin_layout Plain Layout + +(0.046038860904377095, 1.2008630269232545e-11, 262145) +\end_layout + +\begin_layout Plain Layout + +(0.046038860123720424, -3.0025314137784918e-12, 524289) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename qromb_errvscalls.png + +\end_inset + + +\end_layout + +\begin_layout Standard +This plot shows that the error drops by an order of magnitude (log-log plot) + with an order of magnitude increase in number of function calls. +\end_layout + +\begin_layout Subsection +Split integral into (0,1) and (1,20) parts +\end_layout + +\begin_layout Standard +On splitting the integral into (0,1) and (1,20) parts, a considerably smaller + number of function calls are required to obtain the same accuracy, and + even far beyond the required accuracy. +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +counts=[] +\end_layout + +\begin_layout Plain Layout + +errors=[] +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +for i in range(-1,-11,-1): +\end_layout + +\begin_layout Plain Layout + + x1=r.qromb(integrand,0,1,10**i) +\end_layout + +\begin_layout Plain Layout + + x2=r.qromb(integrand,1,20,10**i) +\end_layout + +\begin_layout Plain Layout + + print x1[0]+x2[0],x1[0]+x2[0]-exact(),x1[2]+x2[2] +\end_layout + +\begin_layout Plain Layout + + counts.append(x1[2]+x2[2]) +\end_layout + +\begin_layout Plain Layout + + errors.append(abs(x1[0]+x2[0]-exact())) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +loglog(counts, errors) +\end_layout + +\begin_layout Plain Layout + +title("Error vs number of function calls (split 0-1 and 1-20)") +\end_layout + +\begin_layout Plain Layout + +xlabel("Number of function calls") +\end_layout + +\begin_layout Plain Layout + +ylabel("Error in integration") +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename qromb_errvscalls_split.png + +\end_inset + + +\end_layout + +\begin_layout Standard +The plot above has a much steeper slope as compared to the naive Romberg + integration being applied on the full interval (0,20). + This is a result of correctly using the information of the sharp peak at + x=1 for splitting the integral to obtain a more precise numerical solution. + The comparison on the same plot is given below: +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename comparison.png + +\end_inset + + +\end_layout + +\begin_layout Subsection +Implementation of qromb using trapzd and polint +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +import romberg as r +\end_layout + +\begin_layout Plain Layout + +global count=0 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def func(u): +\end_layout + +\begin_layout Plain Layout + + return 2*u*sp.jv(3,2.7*u)**2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def func2(u): +\end_layout + +\begin_layout Plain Layout + + return 2*sp.kv(3,1.2*u)**2*u*abs(sp.jv(3,2.7)/sp.kv(3,1.2))**2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def integrand(u): +\end_layout + +\begin_layout Plain Layout + + global count; count+=1; +\end_layout + +\begin_layout Plain Layout + + if u<1.0: +\end_layout + +\begin_layout Plain Layout + + return func(u) +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + return func2(u) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +order=5 +\end_layout + +\begin_layout Plain Layout + +s_out=0.0 +\end_layout + +\begin_layout Plain Layout + +xx=[]; yy=[] //polynomial is a function of h^2 and not h +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +for i in range(1,order+1): +\end_layout + +\begin_layout Plain Layout + + s_out=r.trapzd(integrand,0,20,s_out,i) +\end_layout + +\begin_layout Plain Layout + + xx.append((20.0/(2**(i-1)))**2) +\end_layout + +\begin_layout Plain Layout + + yy.append(s_out) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +y,err=r.polint(xx,yy,0) //extrapolation to h=0 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Varying order of qromb vs number of function calls +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename rombergCalls.png + +\end_inset + + +\end_layout + +\begin_layout Standard +As can be seen from the semilog plot above, the number of calls decreases + from n=5 to n=11, then increases beyond that, for a fixed tolerance of + +\begin_inset Formula $10^{-8}$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Spline Integration using B-Spline from scipy +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename splineErr.png + +\end_inset + + +\begin_inset Newline newline +\end_inset + +This plot shows that the error drops by an order of magnitude (log-log plot) + with an order of magnitude increase in number of function calls. + Spline interpolation of the function forces it to consider the kink at + x=1 as an n-th order smooth polynomial, thus reducing the accuracy of the + fit. + As more points are included, the value around x=1 is sampled even more + closely, thus improving the accuracy, but not actually addressing the problem, + that is the kink at x=1. +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +In : for i in range(4,20): +\end_layout + +\begin_layout Plain Layout + + .....: x=linspace(0,20,2**i) +\end_layout + +\begin_layout Plain Layout + + .....: y=integrand(x) +\end_layout + +\begin_layout Plain Layout + + .....: tck=si.splrep(x,y) +\end_layout + +\begin_layout Plain Layout + + .....: I=si.splint(0,20,tck) +\end_layout + +\begin_layout Plain Layout + + .....: print I-exact() +\end_layout + +\begin_layout Plain Layout + + .....: err.append(abs(I-exact())) +\end_layout + +\begin_layout Plain Layout + + .....: +\end_layout + +\begin_layout Plain Layout + +Out: +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +-0.00469338304051 +\end_layout + +\begin_layout Plain Layout + +-0.0200338967057 +\end_layout + +\begin_layout Plain Layout + +0.00456726068437 +\end_layout + +\begin_layout Plain Layout + +-0.000966336076695 +\end_layout + +\begin_layout Plain Layout + +-9.77496443497e-05 +\end_layout + +\begin_layout Plain Layout + +-8.51200801124e-05 +\end_layout + +\begin_layout Plain Layout + +1.04082882958e-05 +\end_layout + +\begin_layout Plain Layout + +-3.96607022308e-06 +\end_layout + +\begin_layout Plain Layout + +-3.43103425497e-07 +\end_layout + +\begin_layout Plain Layout + +-3.30422590164e-07 +\end_layout + +\begin_layout Plain Layout + +4.00474137047e-08 +\end_layout + +\begin_layout Plain Layout + +-1.55316829675e-08 +\end_layout + +\begin_layout Plain Layout + +-1.33063099428e-09 +\end_layout + +\begin_layout Plain Layout + +-1.29013279226e-09 +\end_layout + +\begin_layout Plain Layout + +1.56270989204e-10 +\end_layout + +\begin_layout Plain Layout + +-6.07057945579e-11 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection +Split into two parts (0,1) and (1,20) and then use splint +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +for i in range(4,20): +\end_layout + +\begin_layout Plain Layout + + x1=linspace(0,1,2**(i-1)) +\end_layout + +\begin_layout Plain Layout + + y1=integrand(x1) +\end_layout + +\begin_layout Plain Layout + + tck1=si.splrep(x1,y1) +\end_layout + +\begin_layout Plain Layout + + I1=si.splint(0,1,tck1) +\end_layout + +\begin_layout Plain Layout + + x2=linspace(1,20,2**(i-1)) +\end_layout + +\begin_layout Plain Layout + + y2=integrand(x2) +\end_layout + +\begin_layout Plain Layout + + tck2=si.splrep(x2,y2) +\end_layout + +\begin_layout Plain Layout + + I2=si.splint(1,20,tck2) +\end_layout + +\begin_layout Plain Layout + + print I1+I2-exact() +\end_layout + +\begin_layout Plain Layout + + err.append(abs(I1+I2-exact())) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The above code snippet splits the function into two parts, one before 1 + and one after 1. + The number of points for a given evaluation remains the same as before + (half the points are used before 1, and half are used after 1: this could + have been made as 1/20 before 1 and 19/20 after 1 as well). +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename splineErr_withSplit.png + +\end_inset + + +\end_layout + +\begin_layout Standard +The above graph decays much faster than the previously obtained graph. + This can be seen from the image below. +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename comparison_splint.png + +\end_inset + + +\end_layout + +\begin_layout Standard +The green line is obtained when the function is split over the two intervals, + whereas the blue one is obtained when considering a single interval (0,20). + This shows that spline fitting/integration gives a better result when discontin +uities or irregularities/kinks of the function are excluded/split. + This allows us to separately fit the individual halves of the function + and thus obtain a better fit. +\end_layout + +\begin_layout Subsection +Custom 3-point trapzd function for Romberg +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +def trap3(func, a, b, n): +\end_layout + +\begin_layout Plain Layout + + if(n==1): +\end_layout + +\begin_layout Plain Layout + + return 0.5*(b-a)*(func(a)+func(b)) +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + d = (float)(b-a)/3**(n-1) +\end_layout + +\begin_layout Plain Layout + + sum=0.0 +\end_layout + +\begin_layout Plain Layout + + x=a+d +\end_layout + +\begin_layout Plain Layout + + while(xPJFa=nYhH5&D#(f7zIp#90)eh=8A9T1sfxAP_kE$bV5X1v5o`Q>j7b8314NjMScw;OmJP=2ztR9~DvQ;M-Y)NJJ2PTS@ePz^^5+pJMr>yzti~ z6Z6wE#9Gb~6UFiHmLM(eP^yliQ~n=U#3qNJ~#q;hwCFzZJ*<*X=qsO*VU7p z7Kv}(V>HnqUHg;wdVRP^yVCMuOF>{zP^!b$B*7DHB3)hGYP+?zCLav8^HWTGn*OqP zYN)8FyCTndEGGHLo#t^LtL6si6P22cuqqemmYWg#T)&;FlqTiq;-Z|d%}wsI{!}i3 zE6nJ^Ztc%Nt_E9eT^+q*Dte3GZRrC_gXu=EAMNdX$yev?tBY1nC%M={o8vhJ^-?3n zZ{A0P>{dfig9<=52!xz^3ZUs7ZQuVG^=hB)L=45kipI$e5qQIjamlrg+p0MU= zmd(z#eyly;?W(M)VRPCw)GRd-77;-g{dli#gXi*M%R@O;1atLwR!c#{W!$$h4^G$P 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z@bYRqF`Xi*H;_Zj_U+yN4TZbXv9rTqr}^I5pw1?2_vP;S8G>zYD<`<2(iEol<@LWa z6M-kMV=G2wz%Gp0B70Y8nrSy{d#<-klgi*V10vpNH)dU)%yfei?qwm+o$S=;ry=Uh zlx8S{nGKLd%y6M_<%xWsctVPEpmta(K{#$7o%H3&RwOWXdZ98rzk6u49t`_(sC2c0 z?Rc{HpHlPx=byhKgRiwTx%$r^R}>}AYRtD57g - + Romberg Integration (Week 4) @@ -83,6 +83,13 @@

The above plot shows that beyond x=2 (approx), the function behaves like an exponentially decaying function, and by the time x=10, the value has dropped by 10 orders of magnitude. +
+
+Finally, on a log-log scale: +
+
+figure loglog_fnplot.png +
Code: @@ -266,7 +273,7 @@

The plot obtained (log-log plot due to large order variations):

-figure errorVScalls.png +figure errorVScalls.png
@@ -309,7 +316,7 @@

-figure qromb_errvscalls.png +figure qromb_errvscalls.png
@@ -343,14 +350,14 @@

-figure qromb_errvscalls_split.png +figure qromb_errvscalls_split.png
The plot above has a much steeper slope as compared to the naive Romberg integration being applied on the full interval (0,20). This is a result of correctly using the information of the sharp peak at x=1 for splitting the integral to obtain a more precise numerical solution. The comparison on the same plot is given below:
-figure comparison.png +figure comparison.png

@@ -392,7 +399,7 @@

1.9 Varying order of qromb vs number of function calls

-figure rombergCalls.png +figure rombergCalls.png
@@ -402,7 +409,7 @@

1.10 Spline Integration using B-Spline from scipy

-figure splineErr.png +figure splineErr.png
This plot shows that the error drops by an order of magnitude (log-log plot) with an order of magnitude increase in number of function calls. Spline interpolation of the function forces it to consider the kink at x=1 as an n-th order smooth polynomial, thus reducing the accuracy of the fit. As more points are included, the value around x=1 is sampled even more closely, thus improving the accuracy, but not actually addressing the problem, that is the kink at x=1.
@@ -463,14 +470,14 @@

The above code snippet splits the function into two parts, one before 1 and one after 1. The number of points for a given evaluation remains the same as before (half the points are used before 1, and half are used after 1: this could have been made as 1/20 before 1 and 19/20 after 1 as well).

-figure splineErr_withSplit.png +figure splineErr_withSplit.png
The above graph decays much faster than the previously obtained graph. This can be seen from the image below.
-figure comparison_splint.png +figure comparison_splint.png
@@ -520,10 +527,6 @@

Use of the above code snippet gives an error y-exact() as 3.4766x10 − 10, and requires N=265741 calls to the function “integrand”. From section 1.6, it is seen that the qromb function in the romberg module requires 524289 function calls for a similar order tolerance. This is about 2 times more than that required for the 3-point Romberg as programmed above, which does not even consider logic for utilizing previously used points.

- - diff --git a/week3_RombergInterpolation/week3_1.lyx b/week3_RombergInterpolation/week3_1.lyx index a1b603d..388912b 100644 --- a/week3_RombergInterpolation/week3_1.lyx +++ b/week3_RombergInterpolation/week3_1.lyx @@ -1,7 +1,9 @@ -#LyX 2.0 created this file. For more info see http://www.lyx.org/ -\lyxformat 413 +#LyX 2.2 created this file. For more info see http://www.lyx.org/ +\lyxformat 508 \begin_document \begin_header +\save_transient_properties true +\origin unavailable \textclass article \use_default_options true \maintain_unincluded_children false @@ -9,16 +11,16 @@ \language_package default \inputencoding auto \fontencoding global -\font_roman default -\font_sans default -\font_typewriter default +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false -\font_sf_scale 100 -\font_tt_scale 100 - +\font_sf_scale 100 100 +\font_tt_scale 100 100 \graphics default \default_output_format default \output_sync 0 @@ -28,15 +30,24 @@ \use_hyperref false \papersize default \use_geometry false -\use_amsmath 1 -\use_esint 1 -\use_mhchem 1 -\use_mathdots 1 +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 \cite_engine basic +\cite_engine_type default +\biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false +\justification true \use_refstyle 1 \index Index \shortcut idx @@ -210,6 +221,19 @@ The above plot shows that beyond x=2 (approx), the function behaves like dropped by 10 orders of magnitude. \end_layout +\begin_layout Standard +Finally, on a log-log scale: +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename loglog_fnplot.png + +\end_inset + + +\end_layout + \begin_layout Standard Code: \end_layout diff --git a/week7_ProbabilityDistbn/surface3d_demo.py b/week7_ProbabilityDistbn/surface3d_demo.py index b1dd729..5041feb 100644 --- a/week7_ProbabilityDistbn/surface3d_demo.py +++ b/week7_ProbabilityDistbn/surface3d_demo.py @@ -25,7 +25,7 @@ linewidth=0, antialiased=False) # Customize the z 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+\end_layout + +\begin_layout Plain Layout + + return res[0] +\end_layout + +\begin_layout Plain Layout + + iters += 1 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # break after no_improv_break iterations with no improvement +\end_layout + +\begin_layout Plain Layout + + print '...best so far:', best +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + if best < prev_best - no_improve_thr: +\end_layout + +\begin_layout Plain Layout + + no_improv = 0 +\end_layout + +\begin_layout Plain Layout + + prev_best = best +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + no_improv += 1 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + if no_improv >= no_improv_break: +\end_layout + +\begin_layout Plain Layout + + return res[0] +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # centroid +\end_layout + +\begin_layout Plain Layout + + x0 = [0.] * dim +\end_layout + +\begin_layout Plain Layout + + for tup in res[:-1]: +\end_layout + +\begin_layout Plain Layout + + for i, c in enumerate(tup[0]): +\end_layout + +\begin_layout Plain Layout + + x0[i] += c / (len(res)-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # reflection +\end_layout + +\begin_layout Plain Layout + + xr = x0 + alpha*(x0 - res[-1][0]) +\end_layout + +\begin_layout Plain Layout + + rscore = f(xr) +\end_layout + +\begin_layout Plain Layout + + if res[0][1] <= rscore < res[-2][1]: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xr, rscore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # expansion +\end_layout + +\begin_layout Plain Layout + + if rscore < res[0][1]: +\end_layout + +\begin_layout Plain Layout + + xe = x0 + gamma*(x0 - res[-1][0]) +\end_layout + +\begin_layout Plain Layout + + escore = f(xe) +\end_layout + +\begin_layout Plain Layout + + if escore < rscore: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xe, escore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xr, rscore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # contraction +\end_layout + +\begin_layout Plain Layout + + xc = x0 + rho*(x0 - res[-1][0]) +\end_layout + +\begin_layout Plain Layout + + cscore = f(xc) +\end_layout + +\begin_layout Plain Layout + + if cscore < res[-1][1]: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xc, cscore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # reduction +\end_layout + +\begin_layout Plain Layout + + x1 = res[0][0] +\end_layout + +\begin_layout Plain Layout + + nres = [] +\end_layout + +\begin_layout Plain Layout + + for tup in res: +\end_layout + +\begin_layout Plain Layout + + redx = x1 + sigma*(tup[0] - x1) +\end_layout + +\begin_layout Plain Layout + + score = f(redx) +\end_layout + +\begin_layout Plain Layout + + nres.append([redx, score]) +\end_layout + +\begin_layout Plain Layout + + res = nres +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The function definition and sample output is below: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +def f(x): +\end_layout + +\begin_layout Plain Layout + + val=(x[0]-0.5)**2+(0.5*x[0]+x[1]-0.5)**2+(0.25*x[0]+0.5*x[1]+x[2]-0.5)**2 +\end_layout + +\begin_layout Plain Layout + + return 1.5-jn(0,val) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +OUTPUT: +\end_layout + +\begin_layout Plain Layout + +In [153]: nelder_mead(f,np.array([0.,0.,0.])) +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.584621234078 1 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.537307160442 2 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.537307160442 3 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.513054706896 4 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.50040693318 5 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.50040693318 6 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500112477919 7 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500112477919 8 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 9 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 10 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 11 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 12 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 13 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 14 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002195289 15 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002195289 16 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500000221936 17 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500000221936 18 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.50000004938 19 +\end_layout + +\begin_layout Plain Layout + +Out[153]: [array([ 0.50419441, 0.26780545, 0.23450568]), 0.50000004938029519] +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +This shows that the method requires 19 iterations to converge to the minimum + from the origin starting point. +\end_layout + +\begin_layout Subsection +Minimizing along coordinate directions +\end_layout + +\begin_layout Standard +Minimizing along conjugate directions requires a loop in which the best + estimate of the minima is updated by moving along x,y,z respectively. +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +pos=np.array([0,0,0]) +\end_layout + +\begin_layout Plain Layout + +deltax=np.array([1e-4,0,0]) +\end_layout + +\begin_layout Plain Layout + +deltay=np.array([0,1e-4,0]) +\end_layout + +\begin_layout Plain Layout + +deltaz=np.array([0,0,1e-4]) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +for i in range(1000): +\end_layout + +\begin_layout Plain Layout + + ...: prev=f(pos) +\end_layout + +\begin_layout Plain Layout + + ...: if f(pos)>f(pos+deltax): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos+deltax): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos+deltax +\end_layout + +\begin_layout Plain Layout + + ...: elif(f(pos)>f(pos-deltax)): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos-deltax): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos-deltax +\end_layout + +\begin_layout Plain Layout + + ...: if f(pos)>f(pos+deltay): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos+deltay): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos+deltay +\end_layout + +\begin_layout Plain Layout + + ...: elif f(pos)>f(pos-deltay): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos-deltay): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos-deltay +\end_layout + +\begin_layout Plain Layout + + ...: if f(pos)>f(pos+deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos+deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos+deltaz +\end_layout + +\begin_layout Plain Layout + + ...: elif f(pos)>f(pos-deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos-deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos-deltaz +\end_layout + +\begin_layout Plain Layout + + ...: print pos, f(pos) +\end_layout + +\begin_layout Plain Layout + + ...: if abs(prev-f(pos))<1e-10: +\end_layout + +\begin_layout Plain Layout + + ...: break +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +OUTPUT: +\end_layout + +\begin_layout Plain Layout + +[ 0.6667 0.2666 0.2 ] 0.500356779364 +\end_layout + +\begin_layout Plain Layout + +[ 0.5016 0.2692 0.24 ] 0.500000040514 +\end_layout + +\begin_layout Plain Layout + +[ 0.4928 0.2576 0.248 ] 0.500000001151 +\end_layout + +\begin_layout Plain Layout + +[ 0.4968 0.2524 0.2496] 0.50000000003 +\end_layout + +\begin_layout Plain Layout + +[ 0.4989 0.2507 0.2499] 0.5 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Thus, just 5 iterations are required for converging to the minimum in this + case. + This may converge to a different minima depending on the starting point. +\end_layout + +\begin_layout Section +Minimization of difficult functions +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +f(x,y)=u^{2}+v^{2} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\left(\begin{array}{c} +u\\ +v +\end{array}\right)=\left(\begin{array}{cc} +cos\alpha & sin\alpha\\ +-sin\alpha & cos\alpha +\end{array}\right)\left(\begin{array}{c} +x\\ +y +\end{array}\right)-\left(\begin{array}{c} +5\\ +5 +\end{array}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +where +\begin_inset Formula $\alpha=5\left(1.5\sqrt{x^{2}+y^{2}}-1.5\right)$ +\end_inset + +. +\end_layout + +\begin_layout Subsection +Surface plot of the function +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename min2_1.png + +\end_inset + + +\end_layout + +\begin_layout Subsection +Contour plot of the function +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename min2_cnt.png + +\end_inset + + +\end_layout + +\begin_layout Standard +The spiralling manner of the minima is more clearly seen for a larger interval + x,y. +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename min2_cnt2.png + +\end_inset + + +\end_layout + +\begin_layout Standard +Thus, any algorithm that begins at the origin will slowly spiral outward + to find the minima of the function (a value that is strictly positive, + but close to zero). +\end_layout + +\begin_layout Subsection +Downhill Simplex Method (Nelder-Mead) +\end_layout + +\begin_layout Standard +This method will use the simplex algorithm to find the minimum. + [Animation reference: https://stackoverflow.com/questions/42722691/python-matplo +tlib-update-scatter-plot-from-a-function]. + The code for animating it is as follows: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +import matplotlib.pyplot as plt +\end_layout + +\begin_layout Plain Layout + +from matplotlib import cm +\end_layout + +\begin_layout Plain Layout + +from matplotlib.ticker import LinearLocator, FormatStrFormatter +\end_layout + +\begin_layout Plain Layout + +import numpy as np +\end_layout + +\begin_layout Plain Layout + +import mpl_toolkits.mplot3d.axes3d as p3 +\end_layout + +\begin_layout Plain Layout + +from matplotlib import animation +\end_layout + +\begin_layout Plain Layout + +from pylab import * +\end_layout + +\begin_layout Plain Layout + +import copy +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +xpos=[] +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def u(x,y): +\end_layout + +\begin_layout Plain Layout + + alpha=5*(1.5*sqrt(x**2+y**2)-1.5) +\end_layout + +\begin_layout Plain Layout + + u=x*cos(alpha)+y*sin(alpha)-5 +\end_layout + +\begin_layout Plain Layout + + return u +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def v(x,y): +\end_layout + +\begin_layout Plain Layout + + alpha=5*(1.5*sqrt(x**2+y**2)-1.5) + +\end_layout + +\begin_layout Plain Layout + + v=-x*sin(alpha)+y*cos(alpha)-5 +\end_layout + +\begin_layout Plain Layout + + return v +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def f2(x): +\end_layout + +\begin_layout Plain Layout + + return u(x[0],x[1])**2+v(x[0],x[1])**2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +plt.ion() +\end_layout + +\begin_layout Plain Layout + +fig,ax=plt.subplots() +\end_layout + +\begin_layout Plain Layout + +x,y=[],[] +\end_layout + +\begin_layout Plain Layout + +sc=ax.scatter(x,y) +\end_layout + +\begin_layout Plain Layout + +plt.xlim(-5,5) +\end_layout + +\begin_layout Plain Layout + +plt.ylim(-5,5) +\end_layout + +\begin_layout Plain Layout + +plt.draw() +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +def draw(): +\end_layout + +\begin_layout Plain Layout + + global plt +\end_layout + +\begin_layout Plain Layout + + x.append(xpos[0]) +\end_layout + +\begin_layout Plain Layout + + y.append(xpos[1]) +\end_layout + +\begin_layout Plain Layout + + sc.set_offsets(np.c_[x,y]) +\end_layout + +\begin_layout Plain Layout + + fig.canvas.draw_idle() +\end_layout + +\begin_layout Plain Layout + + plt.pause(0.001) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +nelder_mead(f2,np.array([0.0,0.0,0.0])) #function defined above +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The animation goes in a spiralling fashion. + The output at various points is as follows: +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename spiral1.png + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename spiral2.png + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename spiral3.png + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Graphics + filename spiral4.png + +\end_inset + + +\end_layout + +\begin_layout Standard +It is extremely slow... + requiring 9077 iterations to converge to the solution: +\end_layout + +\begin_layout Standard +array([-4.48908229, 5.45022692]), 0.00010828963033755263 +\end_layout + +\begin_layout Subsection +Conjugate Gradient method +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard + +\end_layout + +\end_body +\end_document diff --git a/week8_Minimization/min2asg.lyx~ b/week8_Minimization/min2asg.lyx~ new file mode 100644 index 0000000..b536486 --- /dev/null +++ b/week8_Minimization/min2asg.lyx~ @@ -0,0 +1,969 @@ +#LyX 2.2 created this file. For more info see http://www.lyx.org/ +\lyxformat 508 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language english +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\quotes_language english +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +Minimization in nD +\end_layout + +\begin_layout Author +Rohan Rao (EE14B118) +\end_layout + +\begin_layout Section +Minimization of +\begin_inset Quotes eld +\end_inset + +nice +\begin_inset Quotes erd +\end_inset + + functions +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +f(x,y,z)=1.5-J_{0}\left((x-0.5)^{2}+(0.5x+y-0.5)^{2}+(0.25x+0.5y+z-0.5)^{2}\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +The function minima occurs at (0.5, 0.25, 0.25) since +\begin_inset Formula $J_{0}(x)$ +\end_inset + + is maximum when its real argument is zero. + The initial point is at the origin. +\end_layout + +\begin_layout Subsection +Downhill Simplex (Nelder-Mead) algorithm +\end_layout + +\begin_layout Standard +SOURCE CODE: https://github.com/fchollet/nelder-mead/ +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +def nelder_mead(f, x_start, +\end_layout + +\begin_layout Plain Layout + + step=0.1, no_improve_thr=10e-6, +\end_layout + +\begin_layout Plain Layout + + no_improv_break=10, max_iter=0, +\end_layout + +\begin_layout Plain Layout + + alpha=1., gamma=2., rho=-0.5, sigma=0.5): +\end_layout + +\begin_layout Plain Layout + + ''' +\end_layout + +\begin_layout Plain Layout + + @param f (function): function to optimize, must return a scalar + score +\end_layout + +\begin_layout Plain Layout + + and operate over a numpy array of the same dimensions as x_start +\end_layout + +\begin_layout Plain Layout + + @param x_start (numpy array): initial position +\end_layout + +\begin_layout Plain Layout + + @param step (float): look-around radius in initial step +\end_layout + +\begin_layout Plain Layout + + @no_improv_thr, no_improv_break (float, int): break after no_improv_bre +ak iterations with +\end_layout + +\begin_layout Plain Layout + + an improvement lower than no_improv_thr +\end_layout + +\begin_layout Plain Layout + + @max_iter (int): always break after this number of iterations. +\end_layout + +\begin_layout Plain Layout + + Set it to 0 to loop indefinitely. +\end_layout + +\begin_layout Plain Layout + + @alpha, gamma, rho, sigma (floats): parameters of the algorithm +\end_layout + +\begin_layout Plain Layout + + (see Wikipedia page for reference) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + return: tuple (best parameter array, best score) +\end_layout + +\begin_layout Plain Layout + + ''' +\end_layout + +\begin_layout Plain Layout + + global xpos +\end_layout + +\begin_layout Plain Layout + + # init +\end_layout + +\begin_layout Plain Layout + + dim = len(x_start) +\end_layout + +\begin_layout Plain Layout + + prev_best = f(x_start) +\end_layout + +\begin_layout Plain Layout + + no_improv = 0 +\end_layout + +\begin_layout Plain Layout + + res = [[x_start, prev_best]] +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + for i in range(dim): +\end_layout + +\begin_layout Plain Layout + + x = copy.copy(x_start) +\end_layout + +\begin_layout Plain Layout + + x[i] = x[i] + step +\end_layout + +\begin_layout Plain Layout + + score = f(x) +\end_layout + +\begin_layout Plain Layout + + res.append([x, score]) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # simplex iter +\end_layout + +\begin_layout Plain Layout + + iters = 0 +\end_layout + +\begin_layout Plain Layout + + while 1: +\end_layout + +\begin_layout Plain Layout + + # order +\end_layout + +\begin_layout Plain Layout + + res.sort(key=lambda x: x[1]) +\end_layout + +\begin_layout Plain Layout + + best = res[0][1] +\end_layout + +\begin_layout Plain Layout + + xpos = res[0][0] +\end_layout + +\begin_layout Plain Layout + + draw() +\end_layout + +\begin_layout Plain Layout + + +\end_layout + +\begin_layout Plain Layout + + # break after max_iter +\end_layout + +\begin_layout Plain Layout + + if max_iter and iters >= max_iter: +\end_layout + +\begin_layout Plain Layout + + return res[0] +\end_layout + +\begin_layout Plain Layout + + iters += 1 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # break after no_improv_break iterations with no improvement +\end_layout + +\begin_layout Plain Layout + + print '...best so far:', best +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + if best < prev_best - no_improve_thr: +\end_layout + +\begin_layout Plain Layout + + no_improv = 0 +\end_layout + +\begin_layout Plain Layout + + prev_best = best +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + no_improv += 1 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + if no_improv >= no_improv_break: +\end_layout + +\begin_layout Plain Layout + + return res[0] +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # centroid +\end_layout + +\begin_layout Plain Layout + + x0 = [0.] * dim +\end_layout + +\begin_layout Plain Layout + + for tup in res[:-1]: +\end_layout + +\begin_layout Plain Layout + + for i, c in enumerate(tup[0]): +\end_layout + +\begin_layout Plain Layout + + x0[i] += c / (len(res)-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # reflection +\end_layout + +\begin_layout Plain Layout + + xr = x0 + alpha*(x0 - res[-1][0]) +\end_layout + +\begin_layout Plain Layout + + rscore = f(xr) +\end_layout + +\begin_layout Plain Layout + + if res[0][1] <= rscore < res[-2][1]: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xr, rscore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # expansion +\end_layout + +\begin_layout Plain Layout + + if rscore < res[0][1]: +\end_layout + +\begin_layout Plain Layout + + xe = x0 + gamma*(x0 - res[-1][0]) +\end_layout + +\begin_layout Plain Layout + + escore = f(xe) +\end_layout + +\begin_layout Plain Layout + + if escore < rscore: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xe, escore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + + else: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xr, rscore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # contraction +\end_layout + +\begin_layout Plain Layout + + xc = x0 + rho*(x0 - res[-1][0]) +\end_layout + +\begin_layout Plain Layout + + cscore = f(xc) +\end_layout + +\begin_layout Plain Layout + + if cscore < res[-1][1]: +\end_layout + +\begin_layout Plain Layout + + del res[-1] +\end_layout + +\begin_layout Plain Layout + + res.append([xc, cscore]) +\end_layout + +\begin_layout Plain Layout + + continue +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + + # reduction +\end_layout + +\begin_layout Plain Layout + + x1 = res[0][0] +\end_layout + +\begin_layout Plain Layout + + nres = [] +\end_layout + +\begin_layout Plain Layout + + for tup in res: +\end_layout + +\begin_layout Plain Layout + + redx = x1 + sigma*(tup[0] - x1) +\end_layout + +\begin_layout Plain Layout + + score = f(redx) +\end_layout + +\begin_layout Plain Layout + + nres.append([redx, score]) +\end_layout + +\begin_layout Plain Layout + + res = nres +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The function definition and sample output is below: +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +def f(x): +\end_layout + +\begin_layout Plain Layout + + val=(x[0]-0.5)**2+(0.5*x[0]+x[1]-0.5)**2+(0.25*x[0]+0.5*x[1]+x[2]-0.5)**2 +\end_layout + +\begin_layout Plain Layout + + return 1.5-jn(0,val) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +OUTPUT: +\end_layout + +\begin_layout Plain Layout + +In [153]: nelder_mead(f,np.array([0.,0.,0.])) +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.584621234078 1 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.537307160442 2 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.537307160442 3 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.513054706896 4 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.50040693318 5 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.50040693318 6 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500112477919 7 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500112477919 8 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 9 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 10 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 11 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 12 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 13 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002829454 14 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002195289 15 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500002195289 16 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500000221936 17 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.500000221936 18 +\end_layout + +\begin_layout Plain Layout + +...best so far: 0.50000004938 19 +\end_layout + +\begin_layout Plain Layout + +Out[153]: [array([ 0.50419441, 0.26780545, 0.23450568]), 0.50000004938029519] +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +This shows that the method requires 19 iterations to converge to the minimum + from the origin starting point. +\end_layout + +\begin_layout Subsection +Minimizing along coordinate directions +\end_layout + +\begin_layout Standard +Minimizing along conjugate directions requires a loop in which the best + estimate of the minima is updated by moving along x,y,z respectively. +\end_layout + +\begin_layout Standard +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +pos=np.array([0,0,0]) +\end_layout + +\begin_layout Plain Layout + +deltax=np.array([1e-4,0,0]) +\end_layout + +\begin_layout Plain Layout + +deltay=np.array([0,1e-4,0]) +\end_layout + +\begin_layout Plain Layout + +deltaz=np.array([0,0,1e-4]) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +for i in range(1000): +\end_layout + +\begin_layout Plain Layout + + ...: prev=f(pos) +\end_layout + +\begin_layout Plain Layout + + ...: if f(pos)>f(pos+deltax): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos+deltax): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos+deltax +\end_layout + +\begin_layout Plain Layout + + ...: elif(f(pos)>f(pos-deltax)): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos-deltax): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos-deltax +\end_layout + +\begin_layout Plain Layout + + ...: if f(pos)>f(pos+deltay): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos+deltay): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos+deltay +\end_layout + +\begin_layout Plain Layout + + ...: elif f(pos)>f(pos-deltay): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos-deltay): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos-deltay +\end_layout + +\begin_layout Plain Layout + + ...: if f(pos)>f(pos+deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos+deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos+deltaz +\end_layout + +\begin_layout Plain Layout + + ...: elif f(pos)>f(pos-deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: while f(pos)>f(pos-deltaz): +\end_layout + +\begin_layout Plain Layout + + ...: pos=pos-deltaz +\end_layout + +\begin_layout Plain Layout + + ...: print pos, f(pos) +\end_layout + +\begin_layout Plain Layout + + ...: if abs(prev-f(pos))<1e-10: +\end_layout + +\begin_layout Plain Layout + + ...: break +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +OUTPUT: +\end_layout + +\begin_layout Plain Layout + +[ 0.6667 0.2666 0.2 ] 0.500356779364 +\end_layout + +\begin_layout Plain Layout + +[ 0.5016 0.2692 0.24 ] 0.500000040514 +\end_layout + +\begin_layout Plain Layout + +[ 0.4928 0.2576 0.248 ] 0.500000001151 +\end_layout + +\begin_layout Plain Layout + +[ 0.4968 0.2524 0.2496] 0.50000000003 +\end_layout + +\begin_layout Plain Layout + +[ 0.4989 0.2507 0.2499] 0.5 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Thus, just 5 iterations are required for converging to the minimum in this + case. + This may converge to a different minima depending on the starting point. +\end_layout + +\end_body +\end_document diff --git a/week8_Minimization/min2asg.py b/week8_Minimization/min2asg.py new file mode 100644 index 0000000..36113af --- /dev/null +++ b/week8_Minimization/min2asg.py @@ -0,0 +1,168 @@ +import matplotlib.pyplot as plt +from matplotlib import cm +from matplotlib.ticker import LinearLocator, FormatStrFormatter +import numpy as np +import mpl_toolkits.mplot3d.axes3d as p3 +from matplotlib import animation +from pylab import * +import copy + +xpos=[] + +def u(x,y): + alpha=5*(1.5*sqrt(x**2+y**2)-1.5) + u=x*cos(alpha)+y*sin(alpha)-5 + return u + +def v(x,y): + alpha=5*(1.5*sqrt(x**2+y**2)-1.5) + v=-x*sin(alpha)+y*cos(alpha)-5 + return v + +def f2(x): + return u(x[0],x[1])**2+v(x[0],x[1])**2 + +plt.ion() +fig,ax=plt.subplots() +"""x=linspace(-2,2,1000) +y=linspace(-2,2,1000) +X,Y=meshgrid(x,y) +Z=f2([X,Y]) +cp=plt.contourf(X,Y,Z) +colorbar(cp)""" + +x,y=[],[] +sc=ax.scatter(x,y) +plt.xlim(-5,5) +plt.ylim(-5,5) +plt.draw() + +def draw(): + global plt + x.append(xpos[0]) + y.append(xpos[1]) + sc.set_offsets(np.c_[x,y]) + fig.canvas.draw_idle() + plt.pause(0.001) + +def nelder_mead(f, x_start, + step=0.1, no_improve_thr=10e-6, + no_improv_break=10, max_iter=0, + alpha=1., gamma=2., rho=-0.5, sigma=0.5): + ''' + @param f (function): function to optimize, must return a scalar score + and operate over a numpy array of the same dimensions as x_start + @param x_start (numpy array): initial position + @param step (float): look-around radius in initial step + @no_improv_thr, no_improv_break (float, int): break after no_improv_break iterations with + an improvement lower than no_improv_thr + @max_iter (int): always break after this number of iterations. + Set it to 0 to loop indefinitely. + @alpha, gamma, rho, sigma (floats): parameters of the algorithm + (see Wikipedia page for reference) + + return: tuple (best parameter array, best score) + ''' + global xpos + # init + dim = len(x_start) + prev_best = f(x_start) + no_improv = 0 + res = [[x_start, prev_best]] + + for i in range(dim): + x = copy.copy(x_start) + x[i] = x[i] + step + score = f(x) + res.append([x, score]) + + # simplex iter + iters = 0 + while 1: + # order + res.sort(key=lambda x: x[1]) + best = res[0][1] + xpos = res[0][0] + draw() + + # break after max_iter + if max_iter and iters >= max_iter: + return res[0] + iters += 1 + + # break after no_improv_break iterations with no improvement + print '...best so far:', best, iters + + if best < prev_best - no_improve_thr: + no_improv = 0 + prev_best = best + else: + no_improv += 1 + + if no_improv >= no_improv_break: + return res[0] + + # centroid + x0 = [0.] * dim + for tup in res[:-1]: + for i, c in enumerate(tup[0]): + x0[i] += c / (len(res)-1) + + # reflection + xr = x0 + alpha*(x0 - res[-1][0]) + rscore = f(xr) + if res[0][1] <= rscore < res[-2][1]: + del res[-1] + res.append([xr, rscore]) + continue + + # expansion + if rscore < res[0][1]: + xe = x0 + gamma*(x0 - res[-1][0]) + escore = f(xe) + if escore < rscore: + del res[-1] + res.append([xe, escore]) + continue + else: + del res[-1] + res.append([xr, rscore]) + continue + + # contraction + xc = x0 + rho*(x0 - res[-1][0]) + cscore = f(xc) + if cscore < res[-1][1]: + del res[-1] + res.append([xc, cscore]) + continue + + # reduction + x1 = res[0][0] + nres = [] + for tup in res: + redx = x1 + sigma*(tup[0] - x1) + score = f(redx) + nres.append([redx, score]) + res = nres + +""" +fig = plt.figure() +ax = p3.Axes3D(fig) + +# Plot the surface. +surf = ax.plot_surface(X, Y, Z, cmap=cm.coolwarm, + linewidth=0, antialiased=False) + +# Customize the z axis. +ax.set_zlim(0,100) +ax.zaxis.set_major_locator(LinearLocator(10)) +ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f')) + +# Add a color bar which maps values to colors. +fig.colorbar(surf, shrink=0.5, aspect=5) + +plt.show() +""" + +nelder_mead(f2,np.array([0.0,0.0])) diff --git a/week8_Minimization/spiral1.png b/week8_Minimization/spiral1.png new file mode 100644 index 0000000000000000000000000000000000000000..f43f717ff1c4c0f1ef15f1d62527b7ef63b92c8e GIT binary patch literal 8470 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