Volume 30 (2008)

Interactive Supplement to the paper

Vol. 30 (2008), pp. 128-143 Richard S. Varga, Amos J. Carpenter, and Bryan W. Lewis: The dynamical motion of the zeros of the partial sums of $e^{z}$, and its relationship to discrepancy theory

The page presents an interactive graphical illustration of the results in the paper. The interactive supplement appears in the figure below. A Java-enabled web browser is required to view and interact with the supplement. The figure displays the zeros of the partial sums $s_{n}(nz)$ for $1 \leq n \leq 200$ and computes the associated discrepancy function values for a given sector. The value of $n$ and the sector can be selected interactively.

The figure consist of three parts:

  1. The large plot displays the zeros of $s_{n}(nz)$ for the selected value of n, the Szegő curve, the selected sector, and a brief statement listing the value of $n$ (the degree), the sector angle, a count of the number of zeros of $s_{n}(nz)$, falling in the sector, and a computed value of the discrepancy function for the given data. The zeros appear as small blue crosses; the Szegő curve appears as a red curve. The selected sector is shaded.
  2. A set of controls for selecting the value of n and a button to clear the “history” plot of discrepancy function values appears just below the main plot.
  3. The small plot at the bottom displays up to the most recent 100 computed discrepancy function values.

The zeros of $s_{n}(nz)$ for each value of $n$ from 1 to 200 were computed a priori with extended-precision arithmetic, and stored as Java double precision floating point numbers with approximately 16 decimal digits of accuracy. All computations are performed by the client Java virtual machine using double-precision arithmetic.

The displayed computed values of the discrepancy function are rounded to four decimal digits. Note that the accuracy of the plot data is limited to the pixel display resolution of about 500x500 pixels. The display resolution affects the ability to select sector angles and may slightly affect the display of the zeros.

Figure 1: Interactive Example.

Using the interactive supplement

  • Select a sector (symmetric about the real axis) by dragging the green squares around the unit circle. The selected sector is shaded.
  • Select a polynomial degree by adjusting the slider value near the bottom of the display, or by directly typing a number between 1 and 200 in the text box near the left-bottom of the display.

The archived, compressed code for this applet be found here: zeros.tar.gz.

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