How to improve the calculation accuracy of Matlab?

Sometimes when it comes to very small value calculation, the calculation accuracy of Matlab would not be enough.

There would be fluctuation in the result.

For instance：

l=4;
l1=l;%Tx Mode
l2=l;%Rx Mode
misalignment = -1:1e-3:1;
result = zeros(1,length(misalignment));
channel_func = @(d) 1./d.*exp(-1j.*2*pi./3e-3.*d);
phi_Tx = (0:8-1).*2.*pi./8;
phi_Rx = phi_Tx;
F_Tx = exp(1j.*l1.*phi_Tx).';
F_Rx = exp(-1j.*l2.*phi_Rx).';
%% For different misalignment, it output different result.
for i=1:length(misalignment)
    distance_fun= @(x,y) sqrt(1e4-2*misalignment(i)/1e2.*cos(x)+2*misalignment(i)/1e2*cos(y)-2e-4*cos(x-y));
    H = channel_func(distance_fun(phi_Tx,phi_Rx.'));
    result(i) = abs(F_Rx.'*H*F_Tx);% The key calculation. How can I improve the accuracy of this matrix multiplication?
end
%% Image
figure(1);
set(0,'defaultfigurecolor','w')
set(gcf,'Position',[100 100 700 600]);
plot(misalignment,abs(result));
grid on;
xlabel('distance/meter');
ylabel('Intensity');

And in theory, this curve should be smooth. I think the fluctuation is caused by accuracy limit of Matlab.

Is there any suggestion? If it's possible, you can modify the code directly.

Any help is appreciated.

[VBBV]

you can use, smoothdata on the resulting matrix

l=4;
l1=l;%Tx Mode
l2=l;%Rx Mode
misalignment = -1:1e-3:1;
result = zeros(1,length(misalignment));
channel_func = @(d) 1./d.*exp(-1j.*2*pi./3e-3.*d);
phi_Tx = (0:8-1).*2.*pi./8

phi_Tx = 1×8

0 0.7854 1.5708 2.3562 3.1416 3.9270 4.7124 5.4978

phi_Rx = phi_Tx

phi_Rx = 1×8

0 0.7854 1.5708 2.3562 3.1416 3.9270 4.7124 5.4978

F_Tx = exp(1j.*l1.*phi_Tx).';
F_Rx = exp(-1j.*l2.*phi_Rx).';
%% For different misalignment, it output different result.
for i=1:length(misalignment)
    distance_fun= @(x,y) sqrt(1e4-2*misalignment(i)/1e2.*cos(x)+2*misalignment(i)/1e2*cos(y)-2e-4*cos(x-y));
    H = channel_func(distance_fun(phi_Tx,phi_Rx.'));
    result(i) = abs(F_Rx.'*H*F_Tx);% The key calculation. How can I improve the accuracy of this matrix multiplication?
end
% smooth data with a suitable distribution Guassian
result1 = smoothdata(result,'gaussian',20);
                                      % try with different values
% smooth data with a suitable distribution SGOLAY
result2 = smoothdata(result,'sgolay');
%% Image
figure(1);
set(0,'defaultfigurecolor','w')
set(gcf,'Position',[100 100 700 600]);
plot(misalignment,abs(result1),misalignment,abs(result2));
grid on;
xlabel('distance/meter');
ylabel('Intensity');
legend('Gaussian','Sgolay filter')

[祥宇 崔]

Thanks!

But I think it would not increase the accuracy.

Anyway, I learned the function smoothdata.

[VBBV]

Calculation accuracy depends on how you set the variable limits used in the program but not using vpa.

l=4;
l1=l;%Tx Mode
l2=l;%Rx Mode
digits(128);

format long
misalignment = (-1:1e-2:1) % 1e-2 to something in finer step   1e-4 or 1e-6 etc

misalignment = 1×201

-1.000000000000000 -0.990000000000000 -0.980000000000000 -0.970000000000000 -0.960000000000000 -0.950000000000000 -0.940000000000000 -0.930000000000000 -0.920000000000000 -0.910000000000000 -0.900000000000000 -0.890000000000000 -0.880000000000000 -0.870000000000000 -0.860000000000000 -0.850000000000000 -0.840000000000000 -0.830000000000000 -0.820000000000000 -0.810000000000000 -0.800000000000000 -0.790000000000000 -0.780000000000000 -0.770000000000000 -0.760000000000000 -0.750000000000000 -0.740000000000000 -0.730000000000000 -0.720000000000000 -0.710000000000000

misalignment = -1:rand(1)*1e-4:1  % this will increase accuracy of dependent variables 

misalignment = 1×76132

-1.000000000000000 -0.999973729598183 -0.999947459196366 -0.999921188794550 -0.999894918392733 -0.999868647990916 -0.999842377589099 -0.999816107187283 -0.999789836785466 -0.999763566383649 -0.999737295981832 -0.999711025580015 -0.999684755178199 -0.999658484776382 -0.999632214374565 -0.999605943972748 -0.999579673570931 -0.999553403169115 -0.999527132767298 -0.999500862365481 -0.999474591963664 -0.999448321561848 -0.999422051160031 -0.999395780758214 -0.999369510356397 -0.999343239954581 -0.999316969552764 -0.999290699150947 -0.999264428749130 -0.999238158347313

misalignment = -1:rand(1)*1e-7:1  % same here 

misalignment = 1×28566322

-1.000000000000000 -0.999999929987486 -0.999999859974973 -0.999999789962459 -0.999999719949946 -0.999999649937432 -0.999999579924919 -0.999999509912405 -0.999999439899891 -0.999999369887378 -0.999999299874864 -0.999999229862351 -0.999999159849837 -0.999999089837324 -0.999999019824810 -0.999998949812297 -0.999998879799783 -0.999998809787269 -0.999998739774756 -0.999998669762242 -0.999998599749729 -0.999998529737215 -0.999998459724701 -0.999998389712188 -0.999998319699674 -0.999998249687161 -0.999998179674647 -0.999998109662134 -0.999998039649620 -0.999997969637107

misalignment = vpa(-1:1e-2:1)  % but not this ... time consuming and boring sometimes

misalignment = 

[祥宇 崔]

Thanks!

But the real problem is that the result of my program is too small to calculate for Matlab under 16bit.

By using vpa, it seems more bits can be used that the result is more accurate. One obvious evidence is that the fluctuation is gone:

[VBBV]

ok, Here is the program execution speed if you use vpa with 8 digits

clearvars, clc
l=4;
l1=l;%Tx Mode
l2=l;%Rx Mode
digits(8);  % using 8 digits
tic
misalignment = vpa(-1:1e-2:1);

and it seems you are using 128 digits !! which probably take even much more time.

[祥宇 崔]

hhhhhh, sure. But I have to take that expense since I need the accurate result.

Answer 1

[祥宇 崔]

By using vpa, I improve the accuracy

l=4;
l1=l;%Tx Mode
l2=l;%Rx Mode
digits(128);
misalignment = vpa(-1:1e-2:1);
result = zeros(1,length(misalignment));
channel_func = @(d) 1./d.*exp(-1j.*2*pi./3e-3.*d);
phi_Tx = (0:8-1).*2.*pi./8;
phi_Rx = phi_Tx;
F_Tx = exp(1j.*l1.*phi_Tx).';
F_Rx = exp(-1j.*l2.*phi_Rx).';
%% For different misalignment, it output different result.
for i=1:length(misalignment)
    distance_fun= @(x,y) sqrt(1e4-2*misalignment(i)/1e2.*cos(x)+2*misalignment(i)/1e2*cos(y)-2e-4*cos(x-y));
    H = channel_func(distance_fun(phi_Tx,phi_Rx.'));
    result(i) = abs(F_Rx.'*H*F_Tx);% The key calculation. How can I improve the accuracy of this matrix multiplication?
end
%% Image
figure(1);
set(0,'defaultfigurecolor','w')
set(gcf,'Position',[100 100 700 600]);
plot(misalignment,abs(result));
grid on;
xlabel('distance/meter');
ylabel('Intensity');