Programming Lab 9B Gabriel’s Horn

Programming Lab 9B
Gabriel’s Horn
Topics: Floating-point instructions, numerical integration, precision.
Prerequisite Reading: Chapters 1-9
Revised: December 26, 2020
Click to download
Lab9B-Main.c
Background1
: Gabriel's horn (also known as Torricelli's trumpet) is a geometric figure formed by taking the graph of 1⁄𝑥 and rotating it about
the x-axis from 𝑥 = 1 to ∞. Calculus can be used to show that the volume
it creates is finite (and equal to 𝜋), and that the surface area is infinite.
This creates an apparent paradox because even though the horn could be
filled with paint, there would never be enough paint to coat the surface.
In this lab, the volume (𝑉) and surface area (𝐴) are computed by numerical integration. Conceptually, the horn is divided into a stack of very thin disks of constant thickness 𝑑𝑥 and diminishing radius 𝑟𝑖
so that the volume of the horn may be computed as the sum of their individual
volumes. (A similar approach is used for the surface area.) The thickness of a disk spans two
successive points on the x-axis; since the actual radius at each value of x differs, the radius of
the disk is approximated as the average of these two values.
Implementation: The limited precision of floating-point arithmetic raises three implementation
issues. (1) First, each of the arithmetic operations may produce an imprecise result; this introduces small errors that accumulate during the summation. Accuracy may be improved however,
by removing the common parameters 𝜋 and 𝑑𝑥 from each term and performing these multiplications only once after the
summation is completed. (2) Second, if 𝑑𝑥 is too small, more error will accumulate due to the larger number of terms, but
if 𝑑𝑥 is too large, the approximation of the radius as an average becomes less accurate. The best value of 𝑑𝑥 for this application was found experimentally to be 0.08420836. (3) Third, the diminishing amounts added to the sum eventually become
so small relative to the sum that the sum no longer changes and thus determines when the summation should be terminated.
Assignment: The main program includes a C function named Integrate that
performs the numerical integration. I.e., you can compile and run the program
without writing any assembly. However, your task is to create an equivalent
replacement for Integrate in assembly. The original C version has been defined as “weak”, so that the linker will automatically replace it in the executable
image by the one you create in assembly; you do not need to remove the C version.
Note that the Integrate function must call two other functions (DeltaX and
UpdateDisplay) that are provided in the main program; do not recreate these
in assembly. If your code is correct, the display should look similar to the image
shown. The display is updated as the integration proceeds. The volume (𝑉)
should converge to 𝜋, but the surface area (𝐴) should continue to grow until the
process terminates when the volume stops changing. Holding down the blue
pushbutton will pause the process; releasing it allows the process to resume.
Once the process completes, pressing and releasing the blue pushbutton will
restart the process.
1 https://en.wikipedia.org/wiki/Gabriel%27s_Horn
𝑉 = ∑(𝜋𝑟𝑖
2
) 𝑑𝑥
𝐴 = ∑(2𝜋𝑟𝑖) 𝑑𝑥
𝑟𝑖 = (
1
𝑥𝑖
+
1
𝑥𝑖+1
)/2