The sections that follow explain how to use the function fixpt_look1_func_approx
to create lookup tables. It gives examples that show
how to create lookup tables for the function sin(2πx) on the interval from 0 to 0.25.
To use the function fixpt_look1_func_approx
, you must first
define its parameters. The required parameters for the function are:
funcstr
— Ideal function
xmin
— Minimum input of interest
xmax
— Maximum input of interest
xdt
— x data type
xscale
— x data scaling
ydt
— y data type
yscale
— y data scaling
rndmeth
— Rounding method
In addition there are three optional parameters:
errmax
— Maximum allowed error of the lookup table
nptsmax
— Maximum number of points of the lookup
table
spacing
— Spacing allowed between breakpoints
You must use at least one of the parameters errmax
and
nptsmax
. The next section, Setting Function Parameters for the Lookup Table, gives typical settings for
these parameters.
If you use only the errmax
parameter, without
nptsmax
, the function creates a lookup table with the fewest points,
for which the worstcase error is at most errmax
. See Using errmax with Unrestricted Spacing.
If you use only the nptsmax
parameter without
errmax
, the function creates a lookup table with at most
nptsmax
points, which has the smallest worse case error. See Using nptsmax with Unrestricted Spacing.
The section Specifying Both errmax and nptsmax describes how the
function behaves when you specify both errmax
and
nptsmax
.
You can use the optional spacing
parameter to restrict the spacing
between breakpoints of the lookup table. The options are
'unrestricted'
— Default.
'even'
— Distance between any two adjacent breakpoints
is the same.
'pow2'
— Distance between any two adjacent breakpoints
is the same and the distance is a power of two.
The section Restricting the Spacing and the examples that follow it
explain how to use the spacing
parameter.
To do the examples in this section, you must first set parameter values for the
fixpt_look1_func_approx
function. To do so,
type the following at the MATLAB^{®} prompt:
funcstr = 'sin(2*pi*x)'; % Define the sine function xmin = 0; % Set the minimum input of interest xmax = 0.25; % Set the maximum input of interest xdt = ufix(16); % Set the x data type xscale = 2^16; % Set the x data scaling ydt = sfix(16); % Set the y data type yscale = 2^14; % Set the y data scaling rndmeth = 'Floor'; % Set the rounding method errmax = 2^10; % Set the maximum allowed error nptsmax = 21; % Specify the maximum number of points
If you exit the MATLAB software after typing these commands, you must retype them before trying any of the other examples in this section.
The first example shows how to create a lookup table that has the fewest data points for a specified worstcase error, with unrestricted spacing. Before trying the example, enter the same parameter values given in the section Setting Function Parameters for the Lookup Table, if you have not already done so in this MATLAB session.
Specify the maximum allowed error by typing
errmax = 2^10;
To create the lookup table, type
[xdata, ydata, errworst] = fixpt_look1_func_approx(funcstr, ...
xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[]);
Note that the nptsmax
and spacing
parameters are
not specified.
The function returns three variables:
xdata
— Vector of breakpoints of the lookup table
ydata
— Vector found by applying ideal function sin(2πx) to xdata
errworst
— Specifies the maximum possible error in the
lookup table
The value of errworst
is less than or equal to the value of
errmax
.
You can find the number of X data points by typing
length(xdata)
ans = 16
This means that 16 points are required to approximate sin(2πx) to within the tolerance specified by errmax
.
You can display the maximum error by typing
errworst
errworst = 9.7656e04
You can plot the output of the function fixpt_look1_func_plot
by typing
fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ...
xscale,ydt,yscale,rndmeth);
The upper plot shows the ideal function sin(2πx) and the fixedpoint lookup approximation between the breakpoints. In this example, the ideal function and the approximation are so close together that the two graphs appear to coincide. The lower plot displays the errors.
In this example, the Y data points, returned by the function fixpt_look1_func_approx
as ydata
, are equal to the ideal
function applied to the points in xdata
. However, you can define a
different set of values for ydata
after running fixpt_look1_func_plot
. This can sometimes reduce the maximum error.
You can also change the values of xmin
and xmax
to evaluate the lookup table on a subset of the original interval.
To find the new maximum error after changing ydata
,
xmin
or xmax
, type
errworst = fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax, ... xdt,xscale,ydt,yscale,rndmeth)
The next example shows how to create a lookup table that minimizes the worstcase error for a specified maximum number of data points, with unrestricted spacing. Before starting the example, enter the same parameter values given in the section Setting Function Parameters for the Lookup Table, if you have not already done so in this MATLAB session.
Specify the number of breakpoints in the lookup table by typing
nptsmax = 21;
To create the lookup table, type
[xdata, ydata, errworst] = fixpt_look1_func_approx(funcstr, ...
xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,[],nptsmax);
The empty brackets, []
, tell the function to ignore the parameter
errmax
, which is not used in this example. Omitting
errmax
causes the function fixpt_look1_func_approx
to return a lookup table of size specified by
nptsmax
, with the smallest worstcase error.
The function returns a vector xdata
with 21 points. You can find
the maximum error for this set of points by typing errworst
at the
MATLAB prompt.
errworst
errworst = 5.1139e04
To plot the lookup table along with the errors, type
fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ...
xscale,ydt,yscale,rndmeth);
In the previous two examples, the function fixpt_look1_func_approx
creates lookup tables with unrestricted spacing between the breakpoints. You can restrict
the spacing to improve the computational efficiency of the lookup table, using the spacing
parameter.
The options for spacing are
'unrestricted'
— Default.
'even'
— Distance between any two adjacent breakpoints
is the same.
'pow2'
— Distance between any two adjacent breakpoints
is the same and is a power of two.
Both power of two and even spacing increase the computational speed of the lookup
table and use less command readonly memory (ROM). However, specifying either of the
spacing restrictions along with errmax
usually requires more data
points in the lookup table than does unrestricted spacing to achieve the same degree of
accuracy. The section Effects of Spacing on Speed, Error, and Memory Usage
discusses the tradeoffs between different spacing options.
The next example shows how to create a lookup table that has evenly spaced breakpoints and a specified worstcase error. To try the example, you must first enter the parameter values given in the section Setting Function Parameters for the Lookup Table, if you have not already done so in this MATLAB session.
Next, at the MATLAB prompt type
spacing = 'even'; [xdata, ydata, errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[],spacing);
You can find the number of points in the lookup table by typing:
length(xdata)
ans = 20
To plot the lookup table along with the errors, type
fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ...
xscale,ydt,yscale,rndmeth);
The next example shows how to create a lookup table that has evenly spaced breakpoints and minimizes the worstcase error for a specified maximum number of points. To try the example, you must first enter the parameter values given in the section Setting Function Parameters for the Lookup Table, if you have not already done so in this MATLAB session.
Next, at the MATLAB prompt type
spacing = 'even'; [xdata, ydata, errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,[],nptsmax,spacing);
The result requires 21 evenly spaced points to achieve a maximum absolute error of
2^10.2209
.
To plot the lookup table along with the errors, type
fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ...
xscale,ydt,yscale,rndmeth);
The next example shows how to construct a lookup table that has power of two spacing and a specified worstcase error. To try the example, you must first enter the parameter values given in the section Setting Function Parameters for the Lookup Table, if you have not already done so in this MATLAB session.
Next, at the MATLAB prompt type
spacing = 'pow2'; [xdata, ydata, errworst] = ... fixpt_look1_func_approx(funcstr,xmin, ... xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[],spacing);
To find out how many points are in the lookup table, type
length(xdata)
ans = 33
This means that 33 points are required to achieve the worstcase error specified by
errmax
. To verify that these points are evenly spaced, type
widths = diff(xdata)
This generates a vector whose entries are the differences between consecutive points in
xdata
. Every entry of widths
is
2^{7}.
To find the maximum error for the lookup table, type
errworst
errworst = 3.7209e04
This is less than the value of errmax
.
To plot the lookup table data along with the errors, type
fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ...
xscale,ydt,yscale,rndmeth);
The next example shows how to create a lookup table that has power of two spacing and minimizes the worstcase error for a specified maximum number of points. To try the example, you must first enter the parameter values given in the section Setting Function Parameters for the Lookup Table, if you have not already done so in this MATLAB session:
spacing = 'pow2'; [xdata, ydata, errworst] = ... fixpt_look1_func_approx(funcstr,xmin, ... xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[],spacing);
The result requires 17 points to achieve a maximum absolute error of 2^9.6267.
To plot the lookup table along with the errors, type
fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ...
xscale,ydt,yscale,rndmeth);
If you include both the errmax
and the nptsmax
parameters, the function fixpt_look1_func_approx
tries to find a
lookup table with at most nptsmax
data points, whose worstcase error is
at most errmax
. If it can find a lookup table meeting both conditions, it
uses the following order of priority for spacing:
Power of two
Even
Unrestricted
If the function cannot find any lookup table satisfying both conditions, it ignores
nptsmax
and returns a lookup table with unrestricted spacing, whose
worstcase error is at most errmax
. In this case, the function behaves
the same as if the nptsmax
parameter were omitted.
Using the parameters described in the section Setting Function Parameters for the Lookup Table, the following examples
illustrate the results of using different values for nptsmax
when you
enter
[xdata ydata errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,nptsmax);
The results for three different settings for nptsmax
are as
follows:
nptsmax = 33;
— The function creates the lookup table with
33 points having power of two spacing, as in Example 3.
nptsmax = 21;
— Because the errmax
and
nptsmax
conditions cannot be met with power of two spacing, the
function creates the lookup table with 20 points having even spacing, as in Example
5.
nptsmax = 16;
— Because the errmax
and
nptsmax
conditions cannot be met with either power of two or even
spacing, the function creates the lookup table with 16 points having unrestricted
spacing, as in Example 1.
The following table summarizes the results for the examples. When you specify
errmax
, even spacing requires more data points than unrestricted, and
power of two spacing requires more points than even spacing.
Example  Options  Spacing  WorstCase Error  Number of Points in Table 

1 


 16 
2 


 21 
3 


 20 
4 


 21 
5 


 33 
6 


 17 