{"id":1090,"date":"2021-11-13T00:24:08","date_gmt":"2021-11-13T08:24:08","guid":{"rendered":"https:\/\/gantovnik.com\/bio-tips\/?p=1090"},"modified":"2021-11-13T00:27:57","modified_gmt":"2021-11-13T08:27:57","slug":"189-mandelbrot-set-using-python","status":"publish","type":"post","link":"https:\/\/gantovnik.com\/bio-tips\/2021\/11\/189-mandelbrot-set-using-python\/","title":{"rendered":"#189 Mandelbrot set using python"},"content":{"rendered":"<pre class=\"brush: python; title: ; notranslate\" title=\"\">\r\nimport numpy as np\r\nimport matplotlib.pyplot as plt\r\n# This determines the number of colors used in the plot.\r\n# The larger the value, the longer the script will take.\r\nmax_iterations = 50\r\n# These parameters define the boundaries of the plot\r\nx_min, x_max = -2.5, 1.5\r\ny_min, y_max = -1.5, 1.5\r\n# This parameter controls the grid spacing.  A smaller value gives better\r\n# resolution, but the script will take longer to run.\r\nds = 0.002\r\nX = np.arange(x_min, x_max + ds, ds)\r\nY = np.arange(y_min, y_max + ds, ds)\r\ndata = np.zeros( (X.size, Y.size), dtype='uint')\r\nfor i in range(X.size):\r\n    for j in range(Y.size):\r\n        x0, y0 = X&#x5B;i], Y&#x5B;j]\r\n        x, y = x0, y0\r\n        count = 0\r\n        while count &lt; max_iterations:\r\n            # Update x and y simultaneously.\r\n            x, y = (x0 + x*x - y*y, y0 + 2*x*y)\r\n            # Exit loop if (x,y) is too far from the origin.\r\n            if (x*x + y*y) &gt; 4.0: break\r\n            count += 1\r\n        data&#x5B;i, j] = max_iterations - count\r\nplt.imshow(data.transpose(), interpolation='nearest', cmap='jet')\r\nplt.axis('off')\r\nplt.savefig('ex189.png', dpi=300)\r\nplt.show()\r\n<\/pre>\n<p><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex189.png?resize=1080%2C720&#038;ssl=1\" alt=\"\" width=\"1080\" height=\"720\" class=\"alignnone size-full wp-image-1093\" srcset=\"https:\/\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex189.png 1800w, https:\/\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex189-1280x853.png 1280w, https:\/\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex189-980x653.png 980w, https:\/\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex189-480x320.png 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) and (max-width: 1280px) 1280px, (min-width: 1281px) 1800px, 100vw\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>import numpy as np import matplotlib.pyplot as plt # This determines the number of colors used in the plot. # The larger the value, the longer the script will take. max_iterations = 50 # These parameters define the boundaries of the plot x_min, x_max = -2.5, 1.5 y_min, y_max = -1.5, 1.5 # This parameter [&hellip;]<\/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_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":"","jetpack_post_was_ever_published":false},"categories":[2],"tags":[],"class_list":["post-1090","post","type-post","status-publish","format-standard","hentry","category-python"],"modified_by":"gantovnik","jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p8bH0k-hA","jetpack_likes_enabled":true,"jetpack-related-posts":[{"id":2175,"url":"https:\/\/gantovnik.com\/bio-tips\/2024\/05\/425-an-animation-of-a-bouncing-ball-using-python\/","url_meta":{"origin":1090,"position":0},"title":"#425 An animation of a bouncing ball using python","author":"gantovnik","date":"2024-05-05","format":false,"excerpt":"[code language=\"python\"] import numpy as np import matplotlib.pyplot as plt import matplotlib. Animation as animation # Acceleration due to gravity, m.s-2. g = 9.81 # The maximum x-range of ball's trajectory to plot. XMAX = 5 # The coefficient of restitution for bounces (-v_up\/v_down). cor = 0.65 # The time\u2026","rel":"","context":"In &quot;animation&quot;","block_context":{"text":"animation","link":"https:\/\/gantovnik.com\/bio-tips\/category\/animation\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/05\/ex425-1.gif?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/05\/ex425-1.gif?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/05\/ex425-1.gif?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":1109,"url":"https:\/\/gantovnik.com\/bio-tips\/2021\/11\/193-animation-using-python\/","url_meta":{"origin":1090,"position":1},"title":"#193 Animation using python","author":"gantovnik","date":"2021-11-19","format":false,"excerpt":"[code language=\"python\"] # create an animation import numpy as np import matplotlib.pyplot as plt import matplotlib. Animation as manimation n = 1000 x = np.linspace(0, 6*np.pi, n) y = np.sin(x) # Define the meta data for the movie FFMpegWriter = manimation.writers[\"ffmpeg\"] metadata = dict(title=\"Movie Test\", artist=\"Matplotlib\", comment=\"a red circle following\u2026","rel":"","context":"In &quot;python&quot;","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex193.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex193.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex193.png?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/11\/ex193.png?resize=700%2C400&ssl=1 2x"},"classes":[]},{"id":1746,"url":"https:\/\/gantovnik.com\/bio-tips\/2023\/01\/204-mandelbrot-fractal-using-python-2-2-2-2-2-2-2-2-2-2\/","url_meta":{"origin":1090,"position":2},"title":"#331 Interpolation with Newton&#8217;s polynomial using python","author":"gantovnik","date":"2023-01-04","format":false,"excerpt":"interpolation.py [code language=\"python\"] def interpolation(c,x,x0): # Evaluate Newton's polynomial at x0. # Degree of polynomial n = len(x) - 1 y0 = c[n] for k in range(1,n+1): y0 = c[n-k] + (x0 - x[n-k]) * y0 return y0 def coef(x,y): # Computes the coefficients of Newton's polynomial. # Number of\u2026","rel":"","context":"In &quot;interpolation&quot;","block_context":{"text":"interpolation","link":"https:\/\/gantovnik.com\/bio-tips\/category\/interpolation\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2023\/01\/ex331.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":1984,"url":"https:\/\/gantovnik.com\/bio-tips\/2023\/12\/398-find-points-of-intersection-of-two-circles\/","url_meta":{"origin":1090,"position":3},"title":"#398 Find points of intersection of two circles","author":"gantovnik","date":"2023-12-26","format":false,"excerpt":"- how to find an equation of the common chord of two intersected circles. [code language=\"python\"] import numpy as np import matplotlib.pyplot as plt import math def get_intersections(x0, y0, r0, x1, y1, r1): # circle 1: (x0, y0), radius r0 # circle 2: (x1, y1), radius r1 d=math.sqrt((x1-x0)**2 + (y1-y0)**2)\u2026","rel":"","context":"In &quot;matplotlib&quot;","block_context":{"text":"matplotlib","link":"https:\/\/gantovnik.com\/bio-tips\/category\/matplotlib\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2023\/12\/ex398.png?fit=640%2C480&ssl=1&resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2023\/12\/ex398.png?fit=640%2C480&ssl=1&resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2023\/12\/ex398.png?fit=640%2C480&ssl=1&resize=525%2C300 1.5x"},"classes":[]},{"id":154,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/inexact-solutions-to-odes\/","url_meta":{"origin":1090,"position":4},"title":"Inexact solutions to ODEs","author":"gantovnik","date":"2019-01-08","format":false,"excerpt":"\u00a0 import os import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl import sympy from IPython.display import display sympy.init_printing() mpl.rcParams['text.usetex'] = True import sympy os.chdir(r'D:\\projects\\wordpress\\ex33') os.getcwd() def plot_direction_field(x, y_x, f_xy, x_lim=(-5, 5), y_lim=(-5, 5), ax=None): f_np = sympy.lambdify((x, y_x), f_xy, 'numpy') x_vec = np.linspace(x_lim[0], x_lim[1], 20) y_vec\u2026","rel":"","context":"In &quot;python&quot;","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example33","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example33.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example33.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example33.png?resize=525%2C300 1.5x"},"classes":[]},{"id":2634,"url":"https:\/\/gantovnik.com\/bio-tips\/2024\/07\/436-non-convex-univariate-function-optimization-using-brents-method-in-python\/","url_meta":{"origin":1090,"position":5},"title":"#436 Non-convex univariate function optimization using Brent&#8217;s method in python","author":"gantovnik","date":"2024-07-18","format":false,"excerpt":"Brent\u2019s method is an optimization algorithm that combines a bisecting algorithm (Dekker\u2019s method) and inverse quadratic interpolation. It can be used for constrained and unconstrained univariate function optimization. The Brent-Dekker method is an extension of the bisection method. It is a root-finding algorithm that combines elements of the secant method\u2026","rel":"","context":"In &quot;optimization&quot;","block_context":{"text":"optimization","link":"https:\/\/gantovnik.com\/bio-tips\/category\/optimization\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/07\/ex436.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/07\/ex436.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2024\/07\/ex436.png?resize=525%2C300&ssl=1 1.5x"},"classes":[]}],"_links":{"self":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/1090","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=1090"}],"version-history":[{"count":0,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/1090\/revisions"}],"wp:attachment":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/media?parent=1090"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/categories?post=1090"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/tags?post=1090"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}