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Complex Operations
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Model the Mandelbrot set as a binary indicator function $\chi:\mathbb{C}\in\{0,1\}$, where $\chi(c)=1$ if $c$ belongs to the Mandelbrot set and $\chi(c)=0$ otherwise.
Only consider complex numbers $c=a+ib$ on a discrete grid on a bounded range. It suffices to consider the range $a\in[-2,1]$ and $b\in[-1.2,1.2]$. Furthermore, for efficiency reasons, use a grid spacing that is not too fine. First, try out $\Delta a = \Delta b = 0.01$. To create the grid, you may use the function np.meshgrid.
-Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$). In case that $v_c(K)
-Plot $\chi$ using the function np.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
-
+Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$).
+Plot $\chi$ using the function plt.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
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Complex Operations
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Model the Mandelbrot set as a binary indicator function $\chi:\mathbb{C}\in\{0,1\}$, where $\chi(c)=1$ if $c$ belongs to the Mandelbrot set and $\chi(c)=0$ otherwise.
Only consider complex numbers $c=a+ib$ on a discrete grid on a bounded range. It suffices to consider the range $a\in[-2,1]$ and $b\in[-1.2,1.2]$. Furthermore, for efficiency reasons, use a grid spacing that is not too fine. First, try out $\Delta a = \Delta b = 0.01$. To create the grid, you may use the function np.meshgrid.
-Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$). In case that $v_c(K)
-Plot $\chi$ using the function np.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
-
+Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$).
+Plot $\chi$ using the function plt.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
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Complex Operations
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-
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@@ -13783,9 +13783,9 @@ Exercises and Results
Model the Mandelbrot set as a binary indicator function $\chi:\mathbb{C}\in\{0,1\}$, where $\chi(c)=1$ if $c$ belongs to the Mandelbrot set and $\chi(c)=0$ otherwise.
Only consider complex numbers $c=a+ib$ on a discrete grid on a bounded range. It suffices to consider the range $a\in[-2,1]$ and $b\in[-1.2,1.2]$. Furthermore, for efficiency reasons, use a grid spacing that is not too fine. First, try out $\Delta a = \Delta b = 0.01$. To create the grid, you may use the function np.meshgrid.
-Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$). In case that $v_c(K)
-Plot $\chi$ using the function np.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
-
+Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$).
+Plot $\chi$ using the function plt.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
+
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Exercises and Results
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@@ -13783,9 +13783,9 @@ Exercises and Results
Model the Mandelbrot set as a binary indicator function $\chi:\mathbb{C}\in\{0,1\}$, where $\chi(c)=1$ if $c$ belongs to the Mandelbrot set and $\chi(c)=0$ otherwise.
Only consider complex numbers $c=a+ib$ on a discrete grid on a bounded range. It suffices to consider the range $a\in[-2,1]$ and $b\in[-1.2,1.2]$. Furthermore, for efficiency reasons, use a grid spacing that is not too fine. First, try out $\Delta a = \Delta b = 0.01$. To create the grid, you may use the function np.meshgrid.
-Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$). In case that $v_c(K)
-Plot $\chi$ using the function np.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
-
+Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$).
+Plot $\chi$ using the function plt.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
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Exercises and Results
Model the Mandelbrot set as a binary indicator function $\chi:\mathbb{C}\in\{0,1\}$, where $\chi(c)=1$ if $c$ belongs to the Mandelbrot set and $\chi(c)=0$ otherwise.
Only consider complex numbers $c=a+ib$ on a discrete grid on a bounded range. It suffices to consider the range $a\in[-2,1]$ and $b\in[-1.2,1.2]$. Furthermore, for efficiency reasons, use a grid spacing that is not too fine. First, try out $\Delta a = \Delta b = 0.01$. To create the grid, you may use the function np.meshgrid.
-Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$). In case that $v_c(K)
-Plot $\chi$ using the function np.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
-
+Test for each $c=a+ib$ on that grid, if $(v_c(k))_{k\in\mathbb{N}}$ remains bounded or not. Computationally, this cannot be tested easily. However, usually, the sequence $(v_c(k))$ increases in an exponential fashion in the case that it is not bounded. Therefore, a pragmatic (yet not always correct) test is to fix a maximum number of iterations (e.g., $K = 50$) and a threshold (e.g., $L = 100$).
+Plot $\chi$ using the function plt.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
+
@@ -13818,7 +13818,7 @@ Exercises and Results
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@@ -13839,7 +13839,7 @@ Exercises and Results
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np.meshgrid.np.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).plt.imshow, use the colormap 'gray_r'. Furthermore, use the parameter extent to adjust ranges of the horizontal axis $[-2,1]$ (real part) and vertical axis $[-1.2,1.2]$ (imaginary part).
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Exercises and Results
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