Question

The CORDIC (COordinate Rotation DIgital Calculation) algorithm is used by most calculators to evaluate trigonometric and logarithmic functions. An important number in the CORDIC algorithm, called the aggregate constant, is \(\prod_{n=0}^{\infty} \frac{2^{n}}{\sqrt{1+2^{2 n}}},\) where \(\prod_{n=0}^{N} a_{n}\) represents the product \(a_{0} \cdot a_{1} \cdots a_{N}\). This infinite product is the limit of the sequence $$\left\\{\prod_{n=0}^{0} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \cdot \prod_{n=0}^{1} \frac{2^{n}}{\sqrt{1+2^{2 n}}}, \prod_{n=0}^{2} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \ldots .\right\\}.$$ Estimate the value of the aggregate constant.

Short answer

In order to estimate the value of the aggregate constant which is an infinite product of the sequence $\prod_{n=0}^{\infty} \frac{2^{n}}{\sqrt{1+2^{2 n}}},$ we computed the first few partial products and saw that they were converging fairly quickly. By analyzing the convergence of the partial products, we determined that the aggregation constant is approximately 0.709.

Step-by-step solution

Analyze the sequence

The given sequence is $$ \prod_{n=0}^{\infty} \frac{2^{n}}{\sqrt{1+2^{2 n}}}. $$ To estimate the value of the aggregate constant, we will compute partial products of the given sequence. Let's take a look at the first few terms of the sequence. $$ a_{0}=\frac{2^{0}}{\sqrt{1+2^{2 \cdot 0}}}=\frac{1}{\sqrt{1+1}}=\frac{1}{\sqrt{2}} $$ $$ a_{1}=\frac{2^{1}}{\sqrt{1+2^{2 \cdot 1}}}=\frac{2}{\sqrt{1+4}}=\frac{2}{\sqrt{5}} \hfill \\ $$ $$ a_{2}=\frac{2^{2}}{\sqrt{1+2^{2 \cdot 2}}}=\frac{4}{\sqrt{1+16}}=\frac{4}{\sqrt{17}} $$

Compute partial products

Now, let's find the partial products of the first few terms and check their convergence: $$ P_{0}=\prod_{n=0}^{0} a_{n} = a_{0} =\frac{1}{\sqrt{2}} $$ $$ P_{1}=\prod_{n=0}^{1} a_{n} = a_{0} \cdot a_{1} = \frac{1}{\sqrt{2}} \cdot \frac{2}{\sqrt{5}}=\frac{\sqrt{10}}{10} $$ $$ P_{2}=\prod_{n=0}^{2} a_{n} = a_{0} \cdot a_{1} \cdot a_{2} = \frac{\sqrt{10}}{10} \cdot \frac{4}{\sqrt{17}}=\frac{2\sqrt{170}}{850} $$ We can see that the partial product is converging fairly quickly. Let's compute one more term to see if it provides a closer approximation. $$ a_{3}=\frac{2^{3}}{\sqrt{1+2^{2\cdot 3}}}=\frac{8}{\sqrt{1+64}}=\frac{8}{\sqrt{65}} $$ $$ P_{3}=\prod_{n=0}^{3} a_{n} = a_{0} \cdot a_{1} \cdot a_{2} \cdot a_{3} = \frac{2\sqrt{170}}{850} \cdot \frac{8}{\sqrt{65}}=\frac{64\sqrt{2210}}{2210} $$ We've now computed the partial products \(P_{0}, P_{1}, P_{2},\) and \(P_{3}.\) Comparing their values, we see that they are converging fairly quickly: $$ P_{0} \approx 0.707 \\ P_{1} \approx 0.707 \\ P_{2} \approx 0.710 \\ P_{3} \approx 0.709 $$

Estimate the aggregate constant

Since the partial products are converging quickly, we can use \(P_{3}\) as our estimate for the aggregate constant. Therefore, we estimate the aggregate constant to be approximately: $$ \prod_{n=0}^{\infty} \frac{2^{n}}{\sqrt{1+2^{2n}}} \approx 0.709 $$

Key concepts

Aggregate Constant

The aggregate constant is a critical component in algorithms like the CORDIC, which is widely applied in digital computation of trigonometric and logarithmic functions. In this context, the aggregate constant is defined through a mathematical term that involves an infinite product. It is imperative for the constant to be accurately determined as it functions like a scaling factor that can optimize calculations in algorithms.

When we refer to an aggregate constant in the CORDIC algorithm, we're specifically considering:

• The infinite product that forms from a specific sequence of calculations.
• This mathematical construct resides at the core of executing efficient computation steps.
• The aggregate constant for CORDIC helps achieve consistent and reliable outputs across a wide range of calculations.

By understanding the role of the aggregate constant, one is better equipped to appreciate how CORDIC and other similar algorithms function.

Infinite Product

The concept of an infinite product is foundational in the expression of complex sequences. In the specific scenario of the CORDIC algorithm, it extends indefinitely to represent a sequence's cumulative multiplication. This infinite product is constructed by multiplying an endless series of terms, typically represented mathematically as:

\[\prod_{n=0}^{\infty} a_{n}\]

• An infinite product consists of terms that are usually derived from a calculated sequence.
• The convergence or divergence of these products informs us if we can compute a finite result or if the product is unbounded.
• For applications like the CORDIC algorithm, convergence is vital to ensure meaningful and usable computational results.

Ultimately, despite the intimidating nature of infinite products, their logical progression can help enhance computation frameworks and offer insights into their underlying operations.

Partial Products

Imagine each term of an infinite product as a building block towards a much larger construction, with partial products being each floor laid on this structure. Partial products are essentially the finite multipliers that form part of the infinite product sequence. These products are computed at each step to progressively approximate the infinite product.

Consider any given infinite sequence: the first few terms are multiplied to get the initial partial product, and as more terms are introduced, new partial products are formed:

• Partial products provide insight into the behavior of the sequence.
• They are instrumental in examining convergence; if they stabilize or approach a particular value, the infinite product might have a finite limit.
• These intermediate steps inform the accuracy and speed of convergence.

In the case of the CORDIC algorithm, analyzing partial products allows us to estimate the aggregate constant effectively, providing a practical implementation of theoretical infinite sequences.

Convergence of Sequences

The notion of convergence in sequences refers to the idea that as you progress further along the sequence, the terms approach a specific value or behavior. In mathematical terms, the convergence of a sequence is crucial for determining if it stabilizes towards a value over time.

In our context:

• Convergence tells us if repeated multiplication, such as in partial products, leads to a limited and predictable result.
• It's essential for ensuring the infinite product has a finite result, crucial for algorithms like CORDIC.
• A sequence that converges towards a specific value provides reliability and stability in calculations.

Observing the convergence of partial products estimated earlier shows the sequence achieving a near-constant value, suggesting the infinite product’s finite outcome. Such convergence characterizes efficient computations in digital algorithms.