The Language of Technical Computing
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
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
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