From 78fdbaa3d4ab563a600584006820871a5e377d40 Mon Sep 17 00:00:00 2001 From: Yigitcan Oezer Date: Wed, 4 Oct 2023 12:35:10 +0200 Subject: [PATCH] html file update --- PCP_06_complex.html | 30 +++++++++++++++--------------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/PCP_06_complex.html b/PCP_06_complex.html index 1592320..0517347 100644 --- a/PCP_06_complex.html +++ b/PCP_06_complex.html @@ -13263,7 +13263,7 @@

Basic Definitions
-No description has been provided for this image +No description has been provided for this image
@@ -13361,7 +13361,7 @@

Complex Operations
-No description has been provided for this image +No description has been provided for this image
@@ -13403,7 +13403,7 @@

Complex Operations
-No description has been provided for this image +No description has been provided for this image
@@ -13447,7 +13447,7 @@

Complex Operations
-No description has been provided for this image +No description has been provided for this image
@@ -13505,7 +13505,7 @@

Complex Operations
-No description has been provided for this image +No description has been provided for this image
@@ -13543,7 +13543,7 @@

Complex Operations
-No description has been provided for this image +No description has been provided for this image
@@ -13609,7 +13609,7 @@

Polar Coordinate Plot
-No description has been provided for this image +No description has been provided for this image
@@ -13677,13 +13677,13 @@

Exercises and Results
-No description has been provided for this image +No description has been provided for this image
-No description has been provided for this image +No description has been provided for this image
@@ -13736,7 +13736,7 @@

Exercises and Results
-No description has been provided for this image +No description has been provided for this image
@@ -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).
    • + @@ -13818,7 +13818,7 @@

    Exercises and Results
    -No description has been provided for this image +No description has been provided for this image
    @@ -13839,7 +13839,7 @@

    Exercises and Results
    -No description has been provided for this image +No description has been provided for this image