Open up any linear algebra textbook and have a look at the matrix
entries. Are there all integers? Are they all written as decimals? There's
probably at least some that are fractions. Matrices in computer programs are
almost always in decimal form. The exception is symbolic mathematics programs
like Maple and Mathematica. That's because computers store non-whole numbers as
floating-point values instead.

Floating-points are fine most of the time, but they're often
not exact. What happens if we work the fractions directly?

From there, any calculation using that number is going to propagate and accumulate. With a fraction of integers, you can maintain exactitude longer.

The other possible advantage is speed, (although not in my
implementation, it's R instead of C or assembly code). Computers can add,
subtract, and multiply numbers in binary very efficiently. However, binary
division has to be done with long division* , which is N times slower, where N
is the number of bits of accuracy.

But with fractions, division is just cross-multiplication,
which is two quick multiplications instead of one slow division.

I wrote a set of R functions that handles some common matrix
operations for matrices of fractions instead of floating-point numbers. Only in
edge cases are these preferable than the existing functions, but

The set of R functions in this source file in this google docs folder are a proof of concept. They perform some scalar and matrix operations by
inputting a fraction, outputting a fraction, and at no internal point relying
on floating-point arithmetic.

This isn't trivial. Consider the inner product of
two vectors of length n:

If we write A and C instead as fractions A/B and C/D
respectively, things get messier:

This formula is used in the function innerprod.fracvec(), which takes two vectors of fractions
and returns a single fraction.

Alternatively, one could rewrite matrix operations in terms
of scalar fraction operations instead of trying to do everything in one big
formula, but there's a loss of speed in doing basic arithmetic as function
calls. This is what I ended up doing for fracmat.upper.tri()
the function that converts a square matrix into an upper triangular form.

## Case 1

# Matrix
multiplication, floating point vs fraction matrix

# Make some
random matrices of fractions, and get their product

A =
matrix(sample(1:9),nrow=3,ncol=3)

B =
matrix(sample(1:9),nrow=3,ncol=3)

F =
fracmat(A,B)

C =
matrix(sample(1:9),nrow=3,ncol=3)

D =
matrix(sample(1:9),nrow=3,ncol=3)

G =
fracmat(C,D)

F_float =
A/B

G_float =
C/D

prod_float
= F_float %*% G_float

FG =
multiply.fracmat(F,G)

FG

$`top`

[,1] [,2] [,3]

[1,] 143
97 577

[2,]
2837 23 59

[3,] 172
29 61

$bottom

[,1] [,2] [,3]

[1,] 36
42 126

[2,] 420
15 15

[3,] 7
4 4

prod_frac =
FG$top / FG$bottom

prod_float

[,1] [,2]
[,3]

[1,] 3.972222 2.309524 4.579365

[2,] 6.754762 1.533333 3.933333

[3,]
24.571429 7.250000 15.250000

prod_frac

[,1] [,2]
[,3]

[1,] 3.972222 2.309524 4.579365

[2,] 6.754762 1.533333 3.933333

[3,]
24.571429 7.250000 15.250000

sum(abs(prod_float
- prod_frac))

[1]
1.554312e-15

### Case 2

## Get the
inverse and determinant of a random matrix

## Compare to
the answers from base::det and base::solve

A =
matrix(sample(1:20)[1:9],nrow=3,ncol=3)

B =
matrix(sample(1:20)[1:9],nrow=3,ncol=3)

A

[,1] [,2] [,3]

[1,] 20
18 3

[2,] 2
16 8

[3,] 4
10 7

B

[,1] [,2] [,3]

[1,] 8
9 17

[2,] 10
1 18

[3,] 12
14 5

F =
fracmat(A,B) #Fraction matrix object

F_float =
A/B #Floating point matrix

# Get the
determinants, compare.

fracmat.det(F)

$`top`

[1]
8592592140

$bottom

[1]
159043500

fracmat.det(F)$top
/ fracmat.det(F)$bottom

[1]
54.02668

det(F_float)

[1]
54.02668

# Get the
inverse of each, display, and compare

Finv1 =
fracmat.inverse(F)$top / fracmat.inverse(F)$bottom

Finv2 =
solve(F_float)

Finv1

[,1] [,2] [,3]

[1,] 0.408733982 -0.04949313 -0.03580898

[2,]
-0.002440495 0.06369402 -0.01991270

[3,]
-0.096072464 -0.02071287 0.73297120

Finv2

[,1] [,2] [,3]

[1,] 0.408733982 -0.04949313 -0.03580898

[2,]
-0.002440495 0.06369402 -0.01991270

[3,]
-0.096072464 -0.02071287 0.73297120

sum(abs(Finv1
- Finv2))

[1]
1.065684e-12

Here's a catalogue of the functions included in this
packette. The source file has more detailed comments explaining each one.

These functions establish a new fraction matrix, vector, or
value from a pair of matrices, vectors, or scalars of equal dimensions
respectively (top, bottom). When enforce_integer
= TRUE, the function rounds each entry of the top and bottom parts. The
function fraction() is much
simpler and doesn't include rounding or quality checks because it's only
intended for internal use.

fracmat(top=NA,
bottom=NA, enforce_integer=TRUE)

fracvec(top=NA,
bottom=NA, enforce_integer=TRUE)

fraction(A,B)

These functions convert a fraction matrix or vector back
into a matrix or vector of decimal (floating-point) numbers.

fracvec.to.vec(fv)

fracmat.to.mat(fm)

There are shortcuts to produce commonly used fraction
matrices

allsame.fracmat(top,
bottom=1, nrow, ncol)

allzero.fracmat(nrow,
ncol)

allone.fracmat(nrow,
ncol)

identity.fracmat(size)

Many of the following functions have an argument called
'simplify', which defaults to true. These are functions which call
simplify.fracmat, which divides the numerator and the denominator of each
fraction by their greatest common denominator.

simplify.fracmat(fm,
nPrimes=10, report_overflow = FALSE, fuzzy_reduce = TRUE, maxint = 10^15)

If any part of the fraction matrix is larger than maxint,
which is close to the limit of the modulo function, fuzzy_reduce will scale the
fraction in question down so the larger of the numerator and denominator is
maxint.

fuzzy_reduce(top,
bottom, limit = 10^15)

These do elementwise addition and subtraction of fractions
in two fraction matrices or vectors.

add.fracmat(fracmatAB,
fracmatCD, subtraction = FALSE)

subtract.fracmat(fracmatAB,
fracmatCD)

This does matrix multiplication of fraction matrices. It does
the by iteratively calling the inner product function on rows of the left matrix
(fracmatAB) and columns of the right matrix (fracmatCD)

multiply.fracmat(fracmatAB,
fracmatCD, simplify=TRUE)

innerprod.fracvec(fracvecAB,
fracvecCD, simplify=TRUE)

Just calling a fracmat object will give you two matrices,
but print.fracmat() will output
text of each fraction with the numerator and denominator together.

print.fracmat(fm)

This function converts a single matrix of floats (floatmat)
to a fraction matrix. It sets the denominators to 10^max_digits and the
numerator to floatmat*10^max_digits (it will use fewer digits if exactitude is
reached)

mat.to.fracmat(floatmat,
max_digits = 6, simplify = TRUE)

Does the row operations on a fraction matrix fm to turn it
into an upper triangular form

Returns the upper triangular matrix U and 'inverse', which
records the same row operations starting from the identity

fracmat.upper.tri(fm,
track = FALSE, output_inverse = TRUE)

fracmat.upper.tri.to.ident(fmU,
inv_fm, track = FALSE)

These get the determinant and the inverse of a fraction
matrix. They do so with calls to fracmat.upper.tri
and fracmat.upper.tri.to.ident

fracmat.det(m,
track=FALSE, output_float = FALSE)

fracmat.inverse(m,
track=FALSE)

There is also a decimal (floating-point) matrix version of
these high-level functions, which are useful for developing and testing the
fraction matrix counterparts. float.inverse
and float.det should behave
identically to how base::solve()
and base::det() on single square
matrices.

float.det(m,
track=FALSE)

float.inverse(m,
track=FALSE)

float.upper.tri(m,
track=FALSE, output_inverse =TRUE)

float.upper.tri.to.ident(U,
inv_m, track=FALSE)

These perform elementary fraction arithmetic functions. They
are used internally for row-operation functions like fracmat.upper.tri

negate.fraction(fm,
negate_denom = FALSE)

add.fraction(fracAB,
fracCD)

subtract.fraction(fracAB,
fracCD)

multiply.fraction(fracAB,
fracCD)

divide.fraction(fracAB,
fracCD)

If I come back to this, this is my wish list:

**General development:**

- More unit testing, stress testing, speed testing.

- Further development into a bona-fide package.

- Is any of this paper-worthy?

**Additional matrix functions:**

- Recording of row operations.

- LU decomposition

**Additional presentation:**

- Print out records of matrices after row operations in a
way that Linear Algebra students can use

- Print fraction matrices in latex code

**Avenues for improved fraction simplification:**

- Integration with the big64 package so 64-bit integers can
be handed exactly.

- Possible integration from the VeryLargeIntegers so arbitrarily large integers can be handled, however since VeryLargeIntegers
operates by string manipulation, it is much slower than floating point arithmetic.

- Possibly passing the whole thing over to RcppBigIntAlgos
which is designed to factor large integers.

But what do you think? I am open to suggestion at
jackd@sfu.ca , jack.davis@sportlogiq.com , or @jack_davis_sfu on Twitter.

* (or multiplication by an inverse, or subtraction of
logarithms, which are all either slower or much more memory intensive)

References:

__Linear Algebra: An interactive approach__by Jain and Gunawardena, especially the worked example on page 118 for the detailed test case.__The End of Error: Unum Computing__by John L. Gustafson for its alternative approaches to rounding error.

Shout out to Ashley Lindalia the astrochemist, because 'astrochemist' is an amazing title and because planetary science matters more than anything I do.

## No comments:

## Post a Comment