John R. Cowles

Redundant logarithmic arithmetic

A number system that offers advantages in some situations over conventional floating point and sign/logarithmic number systems is described. Redundant logarithmic arithmetic, like conventional logarithmic arithmetic, relies on table lookups to make the arithmetic unit simpler than an equivalent floating point unit. The cost of 32 bit subtraction in a redundant logarithmic number system is lower than previously published logarithmic subtraction methods. The total memory requirement for a 29-bit redundant logarithmic unit is 16 K words compared to 22 K words by the best previously published conventional sign logarithm unit, assuming similar addition techniques are employed. A redundant logarithmic number system can be implemented with online arithmetic, which would be impractical for a conventional sign logarithm number system. The disadvantages of redundant arithmetic are typical of redundant number systems. First, the redundancy doubles the storage requirements for data values. Second, the representation can become ill-conditioned, especially as a result of iterated multiplications. Third, division and square root operations are more difficult to implement in redundant logarithmic arithmetic. >

Applying features of IEEE 754 to sign/logarithm arithmetic

Various features found in standard floating point arithmetic (IEEE 754) are examined in light of their appropriateness for sign/logarithm arithmetic. The emphasis is on a 32-b word size comparable to IEEE 754 single precision, although other word sizes are possible. A multilayer sign/logarithm format is considered. The lowest layer, similar to previous implementations, would provide only normalized representations but would not provide representations for zero, denormalized values, infinities, and NaNs. The highest layer would provide most of the features found in IEEE 754, including zeros, denormalized values, infinities, and NaNs. Novel algorithms for implementing logarithmic denormalized arithmetic are presented. Simulation results show that the error characteristics of the proposed logarithmic denormalized arithmetic algorithms are similar to those of the denormalized floating point arithmetic in IEEE 754. >

Arithmetic co-transformations in the real and complex logarithmic number systems

The real logarithmic number system, which represents a value with a sign bit and a quantized logarithm, can be generalized to create the complex logarithmic number system, which replaces the sign bit with a quantized angle in a log/polar coordinate system. Although multiplication and related operations are easy in both real and complex systems, addition and subtraction are hard, especially when interpolation is used to implement the system. Both real and complex logarithmic arithmetic benefit from the use of co-transformation, which converts an addition or subtraction from a region where interpolation is expensive to a region where it is easier. Two co-transformations that accomplish this goal are introduced. The first is an approximation based on real analysis of the subtraction logarithm. The second is based on simple algebra that applies for both real and complex values and that works for both addition and subtraction.

Redundant logarithmic number systems

A new number system that offers advantages over conventional floating-point and sign/logarithm number systems is described. Called redundant logarithmic arithmetic, it relies, like conventional logarithmic arithmetic, on table lookups to make the arithmetic unit simpler than an equivalent floating-point unit. The cost of 32-b subtraction in a redundant logarithmic number system is lower than that of previously published logarithmic subtraction methods. Another advantage of a redundant logarithmic number system is that a single arithmetic unit can use the same hardware to add, subtract, or multiply in similar times.<<ETX>>

Error Analysis of the Kmetz/Maenner Algorithm

Use of low-precision logarithms can minimize power consumption and increase the speed of multiply-intensive signal-processing systems, such as FIR filters. Although straight table lookup is the most obvious way to compute the logarithm, Maenner claims to have discovered a technique that produces four extra bits at no cost. We analyze Maenners technique and show that in fact the technique provides only one extra bit of precision. A related technique by Kmetz, which has never been analyzed before, is shown here to be more accurate than Maenners. We compare these techniques to the more complex bipartite technique, and show that Kmetzs technique takes less memory for systems requiring fewer than ten bits of precision.

Fast Fourier Transforms Using the Complex Logarithmic Number System

The complex-logarithmic number system (CLNS), which represents each complex point in log/polar coordinates, may be practical to implement the Fast Fourier Transform (FFT). The roots of unity needed by the FFT have exact representations in CLNS and do not require a ROM.We present an error analysis and simulation results for a radix-two FFT that compares a rectangular fixed-point representation of complex numbers to CLNS. We observe that CLNS saves 9–12 bits in word-size for 256–1024 point FFTs compared to the fixed-point number system while producing comparable accuracy.The consequence of the word-size advantage is that the number of full adders required for CLNS is significantly smaller than for an equivalent fixed-point implementation. The major cost of CLNS is the memory, which unlike conventional LNS, is addressed by both real and imaginary parts. Table-reduction techniques can mitigate this. The simplicity of the CLNS approach requires significantly fewer full adders, which pays for some or all of the extra memory. In applications needing the magnitude of the complex parts, such as a power spectrum, the CLNS approach can actually require less memory than the conventional approach.

Theory Extension in ACL2(r)

ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper presents the logical foundations of ACL2(r). These foundations are also used to justify significant enhancements to ACL2(r).

The least prime quadratic residue and the class number

For an odd prime p, let l(p) denote the least positive prime which is a quadratic residue modulo p. An elementary proof is given of the following: For p ≡ 3 mod 8 and p > 3, l(p) = (p + 1)4 iff the class number h(−p) = 1.

Cluster Definition by the Optimization of Simple Measures

We adopt the following measures of clustering based on simple edge counts in an undirected loop-free graph. Let S be a subset of the points of the graph. The compactness of S is measured by the number of edges connecting points in S to other points in S. The isolation or separation of S is measured by the negative of the number of edges connecting points in S to points not in S. The subset S is a cluster if it is compact and isolated. We study the cluster search problem: find a subset S which maximizes a linear combination of the compactness and (negative) isolation edge counts. We show that a closely related decision problem is NP-complete. We develop a pruned search tree algorithm which is much faster than complete search, especially for graphs which are derived from points embedded in a space of low dimensionality.

Some congruence properties of three well-known sequences: Two notes

Abstract The three sequences mentioned in the title are Ramanujans τ-function, the coefficients c n of Klein, Fricke, and Shimura, and the sequence a n of Apery numbers. In the first note, it is shown that c n ≡ τ ( n )(mod 11). In the second note it is shown that for a prime p , a p +1 ≡ 25 + 60 p (mod p 2 ).