# Integer and Single-Precision Math in MATLAB 7

By Stuart McGarrity, MathWorks

MATLAB 7 now provides integer and single-precision math. This new capability enables processing of integer and single-precision data in its native type, resulting in more efficient memory usage and the ability to process larger, nondouble data sets. This article describes the new MATLAB integer and single-precision math features with application examples for each case.

### Integer Math in MATLAB 7

Before MATLAB 7, all numerical operations in MATLAB were performed in double precision. Double-precision variables, which use 64 bits (8 bytes) of memory storage, accurately represent values to approximately 15 decimal places. Double-precision is a floating-point data type with a large dynamic range.

MATLAB 7 now supports arithmetic on the integers including unsigned integer types uint8, uint16, and uint32, and the signed types int8, int16, and int32. These data types require 8 to 32 bits to be stored in memory. Examples of integer data include output from sensors measuring real-world values, such as audio signals using 8 or 16 bits per sample, or image files using 8 or 24 bits per pixel.

MATLAB supports all arithmetic operators such as +, -, .*, . ⁄, and .^ on these integer data types, as well as some elementary functions such as sum and sort. Other functions that are useful when dealing with integer data types are intmax, intmin, isinteger(x), isa(x,'integer'), cast, ones, zeros, eye and intwarning.

#### The Benefits of Integer Math

Like any computer application, MATLAB has a finite amount of addressable memory to store data. For example, on Windows XP in Release 14 the largest variable MATLAB can store is approximately 1.2GB. By leaving data in its native format and not requiring the conversion to doubles, MATLAB can now handle integer data sets up to eight times larger without running out of memory.

#### Round-off Error and Saturation

Storing real numbers as integers usually results in much larger conversion and arithmetic round-off errors than if they had been stored as doubles. In MATLAB 7 when variables are converted from another data type, the value is rounded to the nearest integer instead of always truncating (as in C), resulting in less round-off error.

To convert a variable to an integer data type, you must pass it to the appropriate function, for example:

>> a=6.4;
>> aint=uint8(a)
aint =
6


Integer data types also have less dynamic range than floating-point data types, therefore the result of an operation can go out of range more easily. In MATLAB 7, if the result of an arithmetic operation is greater that the upper bound of the data type, the variable saturates (it is set to the upper bound). If the result is less than the lower bound of the data type, the variable is set to the lower bound. This is different from C, which wraps around through zero.

When performing arithmetic on integer data types, issues such as round-off error and saturation must be managed by the programmer. One technique is to ensure the data set stays within the data type’s range throughout the execution of the algorithm.

In order to precisely match the overflow and rounding effects that occur in the integer or fixed-point arithmetic of any C compiler or embedded processor, you can use the Fixed-Point Toolbox.

#### Integer Math Rules

You can combine numbers of an integer data type with numbers of the same integer data type or type scalar double. MATLAB performs arithmetic as if both inputs had type double and then converts the result to the same integer data type. The complete set of rules describing how integer types interact with other data is described in the MATLAB documentation.

### Example: Integer math on signal data

The following M-code shows how to perform integer math on some sample signal data. (This section of the article was generated from an M-file using the new MATLAB Editor automatic publishing features.)

#### 1. Load integer signal data

Load a data set comprising signals from four instruments that use 8-bit and 16-bit analog to digital converters. Here we see the data is saved as int8, int16, and uint16, and time is stored as int16.

load integersignal;

% Look at variables
whos Signal1 Signal2 Signal3 Signal4 Time1

Name       Size      Bytes    Class
Signal1    7550x1    7550     int8 array
Signal2    7550x1    7550     int8 array
Signal3    7550x1    15100    uint16 array
Signal4    7550x1    15100    uint16 array
Time1      7550x1    15100    uint16 array


#### 2. Plot integer data

Plot two of the signals to see the signal ranges:

plot(Time1, Signal1, Time1, Signal2);
grid;
legend('Signal1','Signal2');

Click on image to see enlarged view.

Here we see the values are in the range –128 to 127, which is as we would expect for int8. It is likely that these values would need to be scaled to calculate the actual physical value that the signal represents, for example, volts.

#### 3. Process data

Let’s say we want to calculate the sum of Signal1 and Signal2.

SumSig = Signal1 + Signal2; % Here we sum the integer signals.


The resulting variable SumSig is of type int8. We can turn on a warning to alert us if the arithmetic overflows. In the case of int8, overflow will occur if the result of an operation is outside the range –128 to 127. We use the intwarning command to turn on the warning, and re-execute.

intwarning on;
SumSig = Signal1 + Signal2; % Here we sum the integer signals again.

Warning: Out of range value or NaN computed in integer arithmetic.


Performing this checking greatly slows down execution. It is best to turn it on during algorithm development and disable it during final execution to maximize performance.

Plot the sum signal and see where it may have saturated:

intwarning off

plot(Time1, SumSig);
hold on;

% Find where it may have saturated
Saturated = (SumSig == intmin('int8')) | (SumSig == intmax('int8'));

plot(Time1(Saturated),SumSig(Saturated),'rd');grid;
hold off;

Click on image to see enlarged view.

The markers show where the signal may have saturated.

### Example: Integer math on image data

#### 1. Load integer image data

First, load in two JPEG images:

street1=imread('street1.jpg');
whos street1 street2

Name       Size         Bytes     Class
street1    480x640x3    921600    uint8 array
street2    480x640x3    921600    uint8 array


Here we see the images are 24-bit color, stored as three planes of uint8 data.

#### 2. Display images

Next, display the images:

image(street1); % Display image
axis equal; axis off

image(street2); % Display image
axis equal; axis off


#### 3. Scale an image

We can scale an image by a double-precision constant while keeping the result image stored as integers. For example:

duller = 0.5 * street2; % Scale image with a double constant
whos duller

Name      Size         Bytes     Class
duller    480x640x3    921600    uint8 array


Here we display the images side by side.

subplot(1,2,1);
image(street2);
axis off equal tight
title('Original'); % Display image

subplot(1,2,2);
image(duller);
axis off equal tight
title('Duller'); % Display image


Finally, add the two street images together and plot the ghostly result.

combined = street1 + duller; % Add |uint8| images
clf;
image(combined); % Display image
title('Combined');
axis equal; axis off


#### Unsupported Integer Operations

MATLAB 7 does not support complex integer operations and 64-bit integer operations. If you need such features please let us know.

### Single-Precision Math in MATLAB 7

The single-precision data type requires 32 bits to store in memory and is ideal for storing and processing real number values when you don’t require the same level of accuracy as that provided by double precision.

Most of the built-in MATLAB functions that perform mathematical operations on type double now support inputs of type single. This includes arithmetic operators, linear algebra, elementary and trig functions, filtering, and FFTs. Other functions that are useful when dealing with floating-point data types (both single-precision and double-precision) are realmax, realmin, isfloat(x), isa(x,'float'), cast, ones, zeros, eye, inf, nan and eps. The single data type was available in versions before MATLAB 7, however, no mathematical functions or operators to process them were provided.

#### Benefits of Single-Precision Math

By using singles to represent data instead of converting to doubles, you can process single-precision data sets twice the size that could be processed in earlier MATLAB versions. Single-precision values can also simulate the precision used in an embedded system, such as a 32-bit floating-point DSP.

#### Precision Considerations

32-bit single-precision variables represent data to about seven decimal places, which is less accurate than doubles. This inaccuracy results in a greater round-off error in representing values and during arithmetic that compounds during complicated algorithms involving many operations. The programmer should be aware of this issue and address it in his or her program.

To convert a variable to a single data type, you must pass it to the single or cast function:

>> As=single(A) % or
>> As=cast(A,'single');


#### Single-Precision Math Rules

You can combine numbers of type single with numbers of type double or single. MATLAB performs arithmetic as if both inputs had type single and returns a result of type single. (The complete set of rules describing how single types interact with other data is described in the MATLAB documentation.)

### Example: Single-precision math

The following M-code provides examples of arithmetic and linear algebra using single-precision data.

#### Create double-precision data

Create a data set, which is double-precision by default:

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

1    2    0
2    5   -1
4   10   -1


#### Convert to single precision

Convert data to single-precision with the single function.

A = single(Ad);
B = single(rand(3,3));


#### 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 B Ad O Z n

Name    Size      Bytes    Class
A       3x3       36       single array
B       3x3       36       single array
O       1000x1    4000     single array
Z       1000x1    4000     single array
n       1x1       8        double array


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 only 4 bytes (32 bits), whereas doubles require 8 bytes (64 bits).

#### Perform arithmetic and linear algebra

We can perform arithmetic and linear algebra on singles.

C = A .* B % Elementwise arithmetic
C =
0.4103    0.7057     0
1.7873    4.0658    -0.2028
0.2316    0.0986    -0.1987

whos C

Name    Size    Bytes    Class
C       3x3     36       single array


We see the result of this operation, C, is a single. The following operations show more examples of MATLAB processing single-precision data.

C = A * B % Matrix multiplication

C =
2.1976    1.9792    0.5444
5.2309    4.7617    1.0929
10.5197    9.5333    2.3845

A = C / B % Matrix division

A =
1.0000    2.0000    -0.0000
2.0000    5.0000    -1.0000
4.0000   10.0000    -1.0000

E = eig(A) % Eigenvalues

E =
3.7320
0.2680
0.9999

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

F =
7.0000
-2.0000 + 1.7321i
-2.0000 - 1.7321i

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

P =
1    -5    5    -1

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

Q =
1    -10    35    -52    35    -10    1

stem(Q); % Plot the result


In summary, now that MATLAB 7 supports math on integer and single-precision data, you can store such data more efficiently and avoid having to convert it to doubles before processing. You can handle data sets that are twice as large in the case of single-precision data and up to 8 times as large in the case of integer data. You can now stay in the MATLAB environment to perform such work without resorting to C, C++ or Fortran.

Published 2004