The BKM algorithm is a

shift-and-add algorithm Speckle imaging comprises a range of high-resolution astronomical imaging techniques based on the analysis of large numbers of short exposures that freeze the variation of atmospheric turbulence. They can be divided into the shift-and-add ("' ...

for computing

elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...

s, first published in 1994 by Jean-Claude Bajard, Sylvanus Kla, and Jean-Michel Muller. BKM is based on computing complex

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

s (''L-mode'') and

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: * Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value * Ex ...

s (''E-mode'') using a method similar to the algorithm Henry Briggs used to compute logarithms. By using a precomputed table of logarithms of negative powers of two, the BKM algorithm computes elementary functions using only integer add, shift, and compare operations. BKM is similar to

CORDIC CORDIC, short for coordinate rotation digital computer, is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, and exponentials and logarithms In ma ...

, but uses a table of

s rather than a table of

arctangent In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specific ...

s. On each iteration, a choice of coefficient is made from a set of nine complex numbers, 1, 0, −1, i, −i, 1+i, 1−i, −1+i, −1−i, rather than only −1 or +1 as used by CORDIC. BKM provides a simpler method of computing some elementary functions, and unlike CORDIC, BKM needs no result scaling factor. The convergence rate of BKM is approximately one bit per iteration, like CORDIC, but BKM requires more precomputed table elements for the same precision because the table stores logarithms of complex operands. As with other algorithms in the shift-and-add class, BKM is particularly well-suited to hardware implementation. The relative performance of software BKM implementation in comparison to other methods such as

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...

approximations will depend on the availability of fast multi-bit shifts (i.e. a

barrel shifter A barrel shifter is a digital circuit that can bit shift, shift a word (data type), data word by a specified number of bits without the use of any sequential logic, only pure combinational logic, i.e. it inherently provides a binary operation. I ...

) or hardware

floating point In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form ...

arithmetic.

Overview

In order to solve the equation :

\ln(x) = y

the BKM algorithm takes advantage of a basic property of logarithms :

\ln(ab) = \ln(a)+\ln(b)

Using

Pi notation Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a '' product''. Multiplication is often de ...

, this identity generalizes to :

\ln\left(\prod_^n a_k\right) = \sum_^n\ln(a_k)

Because any number can be represented by a product, this allows us to choose any set of values

a_k

which multiply to give the value we started with. In computer systems, it's much faster to multiply and divide by multiples of 2, but because not every number is a multiple of 2, using

a_k = 1+2^m

is a better option than a more simple choice of

a_k = 2^m

. Since we want to start with large changes and get more accurate as

k

increases, we can more specifically use

a_k = 1+2^

, allowing the product to approach any value between 1 and ~4.768, depending on which subset of

a_k

we use in the final product. At this point, the above equation looks like this: :

\ln\left(\prod_ 1+2^\right) = \sum_\ln(1+2^)

This choice of

a_k

reduces the computational complexity of the product from repeated multiplication to simple addition and bit-shifting depending on the implementation. Finally, by storing the values

\ln(1+2^)

in a table, calculating the solution is also a simple matter of addition. Iteratively, this gives us two separate sequences. One sequence approaches the input value

x

while the other approaches the output value

\ln(x)=y

x_k =
   \begin
     1                       & \text k = 0 \\
     x_\cdot (1+2^) & \text x_k \text \leq x \\
     x_                 & \text
   \end

Given this recursive definition and because

x_k

is strictly increasing, it can be shown by induction and

convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...

that :

\lim_x_k = x

for any

1 \leq x \lesssim 4.768

. For calculating the output, we first create the reference table :

A_k = \ln(1+2^)

Then the output is computed iteratively by the definition

y_k =
   \begin
     0             & \text k = 0 \\
     y_ + A_k & \text x_k \text \leq x \\
     y_       & \text
   \end

The conditions in this iteration are the same as the conditions for the input. Similar to the input, this sequence is also strictly increasing, so it can be shown that :

\lim_y_k = y

for any

0 \leq y \lesssim 1.562

. Because the algorithm above calculates both the input and output simultaneously, it's possible to modify it slightly so that

y

is the known value and

x

is the value we want to calculate, thereby calculating the exponential instead of the logarithm. Since x becomes an unknown in this case, the conditional changes from :

\dots \text x_k \text \leq x

to :

\dots \text y_k \text \leq y

Logarithm function

To calculate the logarithm function (L-mode), the algorithm in each iteration tests if

x_n \cdot (1+2^) \le x

. If so, it calculates

x_

and

y_

. After

N

iterations the value of the function is known with an error of

\Delta \ln(x) \le 2^

. Example program for natural logarithm in C++ (see A_e for table): double log_e (const double argument, const int bits = 53) // 1 <= argument <= 4.768462058 Logarithms for bases other than e can be calculated with similar effort. Example program for binary logarithm in C++ (see A_2 for table): double log_2 (const double argument, const int bits = 53) // 1 <= argument <= 4.768462058 The allowed argument range is the same for both examples (1 ≤ Argument ≤ 4.768462058…). In the case of the base-2 logarithm the exponent can be split off in advance (to get the integer part) so that the algorithm can be applied to the remainder (between 1 and 2). Since the argument is smaller than 2.384231…, the iteration of ''k'' can start with 1. Working in either base, the multiplication by s can be replaced with direct modification of the floating point exponent, subtracting 1 from it during each iteration. This results in the algorithm using only addition and no multiplication.

Exponential function

To calculate the exponential function (E-mode), the algorithm in each iteration tests if

y_n + \ln(1+2^) \le y

. If so, it calculates

x_

and

y_

. After

N

iterations the value of the function is known with an error of

\Delta \exp(x) \le 2^

. Example program in C++ (see A_e for table): double exp (const double argument, const int bits = 54) // 0 <= argument <= 1.5620238332

Tables for examples

static const double A_e [] = // A_e[k] = ln (1 + 0.5^k) ; static const double A_2 [] = // A_2[k] = log_2 (1 + 0.5^k) ;

Notes

References