MATLAB

Single Precision Math

This example shows how to perform arithmetic and linear algebra with single precision data. It also shows how the results are computed appropriately in single-precision or double-precision, depending on the input.

Create Double Precision Data

Let's first create some data, which is double precision by default.

Ad = [1 2 0; 2 5 -1; 4 10 -1]
Ad =

     1     2     0
     2     5    -1
     4    10    -1

Convert to Single Precision

We can convert data to single precision with the single function.

A = single(Ad); % or A = cast(Ad,'single');

Create Single Precision Zeros and Ones

We can also create single precision zeros and ones with their respective functions.

n = 1000;
Z = zeros(n,1,'single');
O = ones(n,1,'single');

Let's look at the variables in the workspace.

whos A Ad O Z n
  Name         Size            Bytes  Class     Attributes

  A            3x3                36  single              
  Ad           3x3                72  double              
  O         1000x1              4000  single              
  Z         1000x1              4000  single              
  n            1x1                 8  double              

We can see that some of the variables are of type single and that the variable A (the single precision version of Ad) takes half the number of bytes of memory to store because singles require just four bytes (32-bits), whereas doubles require 8 bytes (64-bits).

Arithmetic and Linear Algebra

We can perform standard arithmetic and linear algebra on singles.

B = A'    % Matrix Transpose
B =

     1     2     4
     2     5    10
     0    -1    -1

whos B
  Name      Size            Bytes  Class     Attributes

  B         3x3                36  single              

We see the result of this operation, B, is a single.

C = A * B % Matrix multiplication
C =

     5    12    24
    12    30    59
    24    59   117

C = A .* B % Elementwise arithmetic
C =

     1     4     0
     4    25   -10
     0   -10     1

X = inv(A) % Matrix inverse
X =

     5     2    -2
    -2    -1     1
     0    -2     1

I = inv(A) * A % Confirm result is identity matrix
I =

     1     0     0
     0     1     0
     0     0     1

I = A \ A  % Better way to do matrix division than inv
I =

     1     0     0
     0     1     0
     0     0     1

E = eig(A) % Eigenvalues
E =

    3.7321
    0.2679
    1.0000

F = fft(A(:,1)) % FFT
F =

   7.0000 + 0.0000i
  -2.0000 + 1.7321i
  -2.0000 - 1.7321i

S = svd(A) % Singular value decomposition
S =

   12.3171
    0.5149
    0.1577

P = round(poly(A)) % The characteristic polynomial of a matrix
P =

     1    -5     5    -1

R = roots(P) % Roots of a polynomial
R =

    3.7321
    1.0000
    0.2679

Q = conv(P,P) % Convolve two vectors
R = conv(P,Q)
Q =

     1   -10    35   -52    35   -10     1


R =

     1   -15    90  -278   480  -480   278   -90    15    -1

stem(R); % Plot the result

A Program that Works for Either Single or Double Precision

Now let's look at a function to compute enough terms in the Fibonacci sequence so the ratio is less than the correct machine epsilon (eps) for datatype single or double.

% How many terms needed to get single precision results?
fibodemo('single')

% How many terms needed to get double precision results?
fibodemo('double')

% Now let's look at the working code.
type fibodemo

% Notice that we initialize several of our variables, |fcurrent|,
% |fnext|, and |goldenMean|, with values that are dependent on the
% input datatype, and the tolerance |tol| depends on that type as
% well.  Single precision requires that we calculate fewer terms than
% the equivalent double precision calculation.
ans =

    19


ans =

    41


function nterms = fibodemo(dtype)
%FIBODEMO Used by SINGLEMATH demo.
% Calculate number of terms in Fibonacci sequence.

% Copyright 1984-2014 The MathWorks, Inc.

fcurrent = ones(dtype);
fnext = fcurrent;
goldenMean = (ones(dtype)+sqrt(5))/2;
tol = eps(goldenMean);
nterms = 2;
while abs(fnext/fcurrent - goldenMean) >= tol
   nterms = nterms + 1;
   temp  = fnext;
   fnext = fnext + fcurrent;
   fcurrent = temp;
end