{"id":197,"date":"2019-01-10T01:43:33","date_gmt":"2019-01-10T09:43:33","guid":{"rendered":"http:\/\/gantovnik.com\/bio-tips\/?p=197"},"modified":"2019-01-10T01:43:33","modified_gmt":"2019-01-10T09:43:33","slug":"julia-fractal","status":"publish","type":"post","link":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/julia-fractal\/","title":{"rendered":"Julia fractal"},"content":{"rendered":"
import os\nimport numba\nimport numpy as np\nimport matplotlib.pyplot as plt\nos.chdir(r'D:\\projects\\wordpress\\ex40')\nos.getcwd()\ndef py_julia_fractal(z_re, z_im, j):\n    for m in range(len(z_re)):\n        for n in range(len(z_im)):\n            z = z_re[m] + 1j * z_im[n]\n            for t in range(256):\n                z = z ** 2 - 0.05 + 0.68j\n                if np.abs(z) > 2.0:\n                    #if (z.real * z.real + z.imag * z.imag) > 4.0: # a bit faster\n                    j[m, n] = t\n                    break\n\njit_julia_fractal = numba.jit(nopython=True)(py_julia_fractal)\nN = 1024\nj = np.zeros((N, N), np.int64)\nz_real = np.linspace(-1.5, 1.5, N)\nz_imag = np.linspace(-1.5, 1.5, N)\njit_julia_fractal(z_real, z_imag, j)\nfig, ax = plt.subplots(figsize=(14, 14))\nax.imshow(j, cmap=plt.cm.RdBu_r,extent=[-1.5, 1.5, -1.5, 1.5])\nax.set_xlabel(\"$\\mathrm{Re}(z)$\", fontsize=18)\nax.set_ylabel(\"$\\mathrm{Im}(z)$\", fontsize=18)\nfig.tight_layout()\nplt.savefig(\"example40.png\", dpi=100)\nplt.show()\nplt.close()\n\n\"example40\"<\/pre>\n","protected":false},"excerpt":{"rendered":"
import os import numba import numpy as np import matplotlib.pyplot as plt os.chdir(r’D:\\projects\\wordpress\\ex40′) os.getcwd() def py_julia_fractal(z_re, z_im, j): for m in range(len(z_re)): for n in range(len(z_im)): z = z_re[m] + 1j * z_im[n] for t in range(256): z = z ** 2 – 0.05 + 0.68j if np.abs(z) > 2.0: #if (z.real * z.real + […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","_lmt_disableupdate":"yes","_lmt_disable":"","jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-197","post","type-post","status-publish","format-standard","hentry","category-python"],"modified_by":null,"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p8bH0k-3b","jetpack_likes_enabled":true,"jetpack-related-posts":[{"id":133,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/spline-interpolation-3\/","url_meta":{"origin":197,"position":0},"title":"#27 Spline interpolation","author":"gantovnik","date":"2019-01-03","format":false,"excerpt":"import os import matplotlib.pyplot as plt import numpy as np from scipy import interpolate os.chdir(r'D:\\data\\scripts\\web1\\ex27') os.getcwd() x = np.array([0, 1, 2, 3, 4, 5, 6, 7]) y = np.array([3, 4, 3.5, 2, 1, 1.5, 1.25, 0.9]) xx = np.linspace(x.min(), x.max(), 100) fig, ax = plt.subplots(figsize=(8, 4)) ax.scatter(x, y) for n\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example27","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example27.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example27.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example27.png?resize=525%2C300 1.5x"},"classes":[]},{"id":139,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/multidimensional-spline\/","url_meta":{"origin":197,"position":1},"title":"Multidimensional Spline","author":"gantovnik","date":"2019-01-04","format":false,"excerpt":"import os import matplotlib.pyplot as plt import numpy as np from scipy import interpolate os.chdir(r'D:\\data\\scripts\\web1\\ex29') os.getcwd() np.random.seed(115925231) x = y = np.linspace(-1, 1, 100) X, Y = np.meshgrid(x, y) def f(x, y): return np.exp(-x**2 - y**2) * np.cos(4*x) * np.sin(6*y) Z = f(X, Y) N = 500 xdata = np.random.uniform(-1,\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example29","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example29.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example29.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example29.png?resize=525%2C300 1.5x"},"classes":[]},{"id":365,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/solution-of-laplace-equation-using-fem\/","url_meta":{"origin":197,"position":2},"title":"#51 Solution of Laplace equation using FEM","author":"gantovnik","date":"2019-01-29","format":false,"excerpt":"","rel":"","context":"In "fem"","block_context":{"text":"fem","link":"https:\/\/gantovnik.com\/bio-tips\/category\/fem\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example61_3.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example61_3.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example61_3.png?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":96,"url":"https:\/\/gantovnik.com\/bio-tips\/2018\/12\/bisection-method\/","url_meta":{"origin":197,"position":3},"title":"#16 Bisection method using python","author":"gantovnik","date":"2018-12-31","format":false,"excerpt":"","rel":"","context":"In "optimization"","block_context":{"text":"optimization","link":"https:\/\/gantovnik.com\/bio-tips\/category\/optimization\/"},"img":{"alt_text":"example16","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example16.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example16.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example16.png?resize=525%2C300 1.5x"},"classes":[]},{"id":93,"url":"https:\/\/gantovnik.com\/bio-tips\/2018\/12\/plots-of-nonlinear-functions\/","url_meta":{"origin":197,"position":4},"title":"Plots of nonlinear functions","author":"gantovnik","date":"2018-12-31","format":false,"excerpt":"import os import matplotlib.pyplot as plt import numpy as np os.chdir('\/home\/vg\/Downloads\/projects\/ex15') os.getcwd() #define model parameters x=np.linspace(-2,2,1000) #examples of nonlinear functions f1=x**2-x-1 f2=x**3-3*np.sin(x) f3=np.exp(x)-2 f4=1-x**2+np.sin(50\/(1+x**2)) fig,axes=plt.subplots(1,4,figsize=(12,3),sharey=True) for n,f in enumerate([f1,f2,f3,f4]): axes[n].plot(x,f,lw=1.5) axes[n].axhline(0,ls=':',color='k') axes[n].set_ylim(-5,5) axes[n].set_xticks([-2,-1,0,1,2]) axes[n].set_xlabel(r'$x$',fontsize=18) axes[0].set_ylabel(r'$f(x)$',fontsize=18) titles=[r'$f(x)=x^2-x-1$',r'$f(x)=x^3-3\\sin(x)$',r'$f(x)=\\exp(x)-2$', r'$f(x)=\\sin\\left(50\/(1+x^2)\\right)+1-x^2$'] for n,title in enumerate(titles): axes[n].set_title(title) plt.savefig(\"example15.png\", dpi=100) plt.show() plt.close()","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example15","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example15.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example15.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example15.png?resize=525%2C300 1.5x"},"classes":[]},{"id":101,"url":"https:\/\/gantovnik.com\/bio-tips\/2018\/12\/solution-of-system-of-nonlinear-equations\/","url_meta":{"origin":197,"position":5},"title":"Solution of system of nonlinear equations","author":"gantovnik","date":"2018-12-31","format":false,"excerpt":"import os import matplotlib.pyplot as plt import numpy as np import scipy os.chdir('\/home\/vg\/Downloads\/projects\/ex17') os.getcwd() def f(x): return [x[1]-x[0]**3-2*x[0]**2+1,x[1]+x[0]**2-1] tol=0.1 a,b=-2,2 x=np.linspace(-3,2,5000) y1=x**3+2*x**2-1 y2=-x**2+1 fig,ax=plt.subplots(figsize=(8,4)) ax.plot(x,y1,'k',lw=1.5) ax.plot(x,y2,'k',lw=1.5) sol1=scipy.optimize.fsolve(f,[-2,2]) sol2=scipy.optimize.fsolve(f,[1,-1]) sol3=scipy.optimize.fsolve(f,[-2,-5]) sols=[sol1,sol2,sol3] colors=['r','b','g'] for idx,s in enumerate(sols): ax.plot(s[0],s[1],colors[idx]+'*',markersize=15) for m in np.linspace(-4,3,80): for n in np.linspace(-20,20,40): x_guess=[m,n] sol=scipy.optimize.fsolve(f,x_guess) idx = (abs(sols-sol)**2).sum(axis=1).argmin() ax.plot(x_guess[0],x_guess[1],colors[idx]+'.')\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example17","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example17.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example17.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2018\/12\/example17.png?resize=525%2C300 1.5x"},"classes":[]}],"_links":{"self":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/197","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/comments?post=197"}],"version-history":[{"count":0,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/197\/revisions"}],"wp:attachment":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/media?parent=197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/categories?post=197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/tags?post=197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}