MLESAC is better than RANSAC by calculating likelihood rather than counting numbers of inliers.
(Torr and Zisserman 2000)
So there is no reason to use RANSAC if we use MLESAC. But when I implied on the plane fitting problem, I got a worse result than RANSAC. It came out similar p_i when I substituted distance errors of each data in equation 19, leading wrong negative log likelihood.
%% MLESAC (REF.PCL)
% data
clc;clear; close all;
f = #(a_hat,b_hat,c_hat,x,y)a_hat.*x+b_hat.*y+c_hat; % z
a = 1;
b = 1;
c = 20;
width = 10;
range = (-width:0.01:width)'; % different from linespace
x = -width+(width-(-width))*rand(length(range),1); % r = a + (b-a).*rand(N,1)
y = -width+(width-(-width))*rand(length(range),1);
X = (-width:0.5:width)';
Y = (-width:0.5:width)';
[X,Y] = meshgrid(X,Y); % for drawing surf
Z = f(a/c,b/c,c/c,X,Y);
z_n = f(a/c,b/c,c/c,x,y); % z/c
% add noise
r = 0.3;
noise = r*randn(size(x));
z_n = z_n + noise;
% add outliers
out_rng = find(y>=8,200);
out_udel = 5;
z_n(out_rng) = z_n(out_rng) + out_udel;
plot3(x,y,z_n,'b.');hold on;
surf(X,Y,Z);hold on;grid on ;axis equal;
p_n = [x y z_n];
num_pt = size(p_n,1);
% compute sigma = median(dist (x - median (x)))
threshold = 0.3; %%%%%%%%% user-defined
medianx = median(p_n(:,1));
mediany = median(p_n(:,2));
medianz = median(p_n(:,3));
medianp = [medianx mediany medianz];
mediadist = median(sqrt(sum((p_n - medianp).*(p_n - medianp),2)));
sigma = mediadist * threshold;
% compute the bounding box diagonal
maxx = max(p_n(:,1));
maxy = max(p_n(:,2));
maxz = max(p_n(:,3));
minx = min(p_n(:,1));
miny = min(p_n(:,2));
minz = min(p_n(:,3));
bound = [maxx maxy maxz]-[minx miny minz];
v = sqrt(sum(bound.*bound,2));
%% iteration
iteration = 0;
num_inlier = 0;
max_iteration = 10000;
max_num_inlier = 0;
k = 1;
s = 5; % number of sample point
probability = 0.99;
d_best_penalty = 100000;
dist_scaling_factor = -1 / (2.0*sigma*sigma);
normalization_factor = 1 / (sqrt(2*pi)*sigma);
Gaussian = #(gamma,disterr,sig)gamma * normalization_factor * exp(disterr.^2*dist_scaling_factor);
Uniform = #(gamma,v)(1-gamma)/v;
while(iteration < k)
% get sample
rand_var = randi([1 length(x)],s,1);
% find coeff. & inlier
A_rand = [p_n(rand_var,1:2) ones(size(rand_var,1),1)];
y_est = p_n(rand_var,3);
Xopt = pinv(A_rand)*y_est;
disterr = abs(sum([p_n(:,1:2) ones(size(p_n,1),1)].*Xopt',2) - p_n(:,3))./sqrt(dot(Xopt',Xopt'));
inlier = find(disterr <= threshold);
outlier = find(disterr >= threshold);
num_inlier = size(inlier,1);
outlier_num = size(outlier,1);
% EM
gamma = 0.5;
iterations_EM = 3;
for i = 1:iterations_EM
% Likelihood of a datam given that it is an inlier
p_i = Gaussian(gamma,disterr,sigma);
% Likelihood of a datum given that it is an outlier
p_o = Uniform(gamma,v);
zi = p_i./(p_i + p_o);
gamma = sum(zi)/num_pt;
end
% Find the log likelihood of the mode -L
d_cur_pentnalty = -sum(log(p_i+p_o));
if(d_cur_pentnalty < d_best_penalty)
d_best_penalty = d_cur_pentnalty;
% record inlier
best_inlier = p_n(inlier,:);
max_num_inlier = num_inlier;
best_model = Xopt;
% Adapt k
w = max_num_inlier / num_pt;
p_no_outliers = 1 - w^s;
k = log(1-probability)/log(p_no_outliers);
end
% RANSAC
% if (num_inlier > max_num_inlier)
% max_num_inlier = num_inlier;
% best_model = Xopt;
%
% % Adapt k
% w = max_num_inlier / num_pt;
% p_no_outliers = 1 - w^s;
% k = log(1-probability)/log(p_no_outliers);
% end
iteration = iteration + 1;
if iteration > max_iteration
break;
end
end
a_est = best_model(1,:);
b_est = best_model(2,:);
c_est = best_model(3,:);
Z_opt = f(a_est,b_est,c_est,X,Y);
new_sur = mesh(X,Y,Z_opt,'edgecolor', 'r','FaceAlpha',0.5); % estimate
title('MLESAC',sprintf('original: a/c = %.2f, b/c = %.2f, c/c = %.2f\n new: a/c = %.2f, b/c = %.2f, c/c = %.2f',a/c,b/c,c/c,a_est,b_est,c_est));
The reference of my source code is from PCL(MLESAC), and I coded it in MATLAB.
I know that lua uses double precision number formats so I wonder if there is a way to convert number directly into string as float value (4 bytes) so that I could send it over udp socket
For Lua 5.3+
local function float32_binary_dump(value)
return ("<f"):pack(value)
end
For Lua 5.1+
local function float32_binary_dump(value)
local img, s, h, d = 2^32 - 2^22, "", 1
if value == value then
img = 2^31 - 2^23
if value < 0 or value == 0 and 1/value < 0 then
value, img = -value, 2^32 - 2^23
end
if value > 0.5 * 2^-149 and value < 2^128 then
-- rounding 64-bit double to 32-bit float
d = math.floor(math.log(value)/math.log(2) + 0.5)
d = value < 2^d and d - 1 or d
local e = 2^(d < -126 and -149 or d - 23)
value = value + 0.5 * e
local r = value % e
value = value - (r == 0 and value % (e + e) or r)
end
-- dumping 32-bit image of float
if value < 2^-149 then
img = img - (2^31 - 2^23)
elseif value <= 2^128 - 2^104 then
if d < -126 then
d, h = -126, 0
end
img = img + (value / 2^d + (d - (-126)) * h - 255) * 2^23
end
end
-- convert 32-bit image to little-endian string
while #s < 4 do
local b = img % 256
s = s..string.char(b)
img = (img - b) / 256
end
return s
end
I'm trying to do a Gaussian bell using the data I am obtaining from a matrix but everytime I try to run the program I obtain this message:
"Error: syntax error, unexpected identifier, expecting end"
The data used to obtain the gaussina bell is a matrix which includes the last point of every n displacements, which are the last position of a particle. I want to know if there is an easier way to obtain the gaussian bell in scilab because I have to also do a fit with an histogram using the same data.
function bla7()
t=4000
n=1000
l=0.067
p=%pi*2
w1=zeros(t,1);
w2=zeros(t,1);
for I=1:t
a=(grand(n,1,"unf",0,p));
x=l*cos(a)
y=l*sin(a)
z1=zeros(n,1);
z2=zeros(n,1);
for i=2:n
z1(i)=z1(i-1)+x(i);
z2(i)=z2(i-1)+y(i);
end
w1(I)=z1($)
w2(I)=z2($)
end
n=10000
w10=zeros(t,1);
w20=zeros(t,1);
for I=1:t
a=(grand(n,1,"unf",0,p));
x=l*cos(a)
y=l*sin(a)
z1=zeros(n,1);
z2=zeros(n,1);
for i=2:n
z1(i)=z1(i-1)+x(i);
z2(i)=z2(i-1)+y(i);
end
w10(I)=z1($)
w20(I)=z2($)
end
n=100
w100=zeros(t,1);
w200=zeros(t,1);
for I=1:t
a=(grand(n,1,"unf",0,p));
x=l*cos(a)
y=l*sin(a)
z1=zeros(n,1);
z2=zeros(n,1);
for i=2:n
z1(i)=z1(i-1)+x(i);
z2(i)=z2(i-1)+y(i);
end
w100(I)=z1($)
w200(I)=z2($)
end
k=70
v=12/k
c1=zeros(k,1)
for r=1:t
c=w1(r)
m=-6+v
n=-6
for g=1:k
if (c<m & c>=n) then
c1(g)=c1(g)+1
m=m+v
n=n+v
else
m=m+v
n=n+v
end
end
end
c2=zeros(k,1)
c2(1)=-6+(6/k)
for b=2:k
c2(b)=c2(b-1)+v
end
y = stdev(w1)
normal1=zeros(k,1)
normal2=zeros(k,1)
bb=-6
bc=-6+v
for wa=1:k
bd=(bb+bc)/2
gauss1=(1/(y*sqrt(2*%pi)))exp(-0.5(bb/y)^2)
gauss2=(1/(y*sqrt(2*%pi)))exp(-0.5(bc/y)^2)
gauss3=(1/(y*sqrt(2*%pi)))exp(-0.5(bd/y)^2)
gauss4=((bc-bb)/6)*(gauss1+gauss2+4*gauss3)
bb=bb+v
bc=bc+v
normal2(wa,1)=gauss4
end
normal3=normal2*4000
k=100
v=24/k
c10=zeros(k,1)
for r=1:t
c=w10(r)
m=-12+v
n=-12
for g=1:k
if (c<m & c>=n) then
c10(g)=c10(g)+1
m=m+v
n=n+v
else
m=m+v
n=n+v
end
end
end
c20=zeros(k,1)
c20(1)=-12+(12/k)
for b=2:k
c20(b)=c20(b-1)+v
end
y = stdev(w10)
normal10=zeros(k,1)
normal20=zeros(k,1)
bb=-12
bc=-12+v
for wa=1:k
bd=(bb+bc)/2
gauss10=(1/(y*sqrt(2*%pi)))exp(-0.5(bb/y)^2)
gauss20=(1/(y*sqrt(2*%pi)))exp(-0.5(bc/y)^2)
gauss30=(1/(y*sqrt(2*%pi)))exp(-0.5(bd/y)^2)
gauss40=((bc-bb)/6)*(gauss10+gauss20+4*gauss30)
bb=bb+v
bc=bc+v
normal20(wa,1)=gauss40
end
normal30=normal20*4000
k=70
v=12/k
c100=zeros(k,1)
for r=1:t
c=w100(r)
m=-6+v
n=-6
for g=1:k
if (c<m & c>=n) then
c100(g)=c100(g)+1
m=m+v
n=n+v
else
m=m+v
n=n+v
end
end
end
c200=zeros(k,1)
c200(1)=-6+(6/k)
for b=2:k
c200(b)=c200(b-1)+v
end
y = stdev(w100)
normal100=zeros(k,1)
normal200=zeros(k,1)
bb=-6
bc=-6+v
for wa=1:k
bd=(bb+bc)/2
gauss100=(1/(y*sqrt(2*%pi)))exp(-0.5(bb/y)^2)
gauss200=(1/(y*sqrt(2*%pi)))exp(-0.5(bc/y)^2)
gauss300=(1/(y*sqrt(2*%pi)))exp(-0.5(bd/y)^2)
gauss400=((bc-bb)/6)*(gauss100+gauss200+4*gauss300)
bb=bb+v
bc=bc+v
normal200(wa,1)=gauss400
end
normal300=normal200*4000
bar(c20,c10,1.0,'white')
plot(c20, normal30, 'b-')
bar(c2,c1,1.0,'white')
plot(c2, normal3, 'r-')
bar(c200,c100,1.0,'white')
plot(c200, normal300, 'm-')
poly1.thickness=3;
xlabel(["x / um"]);
ylabel("molecules");
gcf().axes_size=[500,500]
a=gca();
a.zoom_box=[-12,12;0,600];
a.font_size=4;
a.labels_font_size=5;
a.x_label.font_size = 5;
a.y_label.font_size = 5;
ticks = a.x_ticks
ticks.labels =["-12";"-10";"-8";"-6";"-4";"-2";"0";"2";"4";"6";"8";"10";"12"]
ticks.locations = [-12;-10;-8;-6;-4;-2;0;2;4;6;8;10;12]
a.x_ticks = ticks
endfunction
Each and every one of your gauss variables are missing the multiplication operator in two places. Check every line at it will run. For example, this:
gauss1=(1/(y*sqrt(2*%pi)))exp(-0.5(bb/y)^2)
should be this:
gauss1=(1/(y*sqrt(2*%pi))) * exp(-0.5 * (bb/y)^2)
As for the Gaussian bell, there is no standard function in Scilab. However, you could define a new function to make things more clear in your case:
function x = myGauss(s,b_)
x = (1/(s*sqrt(2*%pi)))*exp(-0.5*(b_/s)^2)
endfunction
Actually, while we're at it, your whole code is really difficult to read. You should define functions instead of repeating code: it helps clarify what you mean, and if there is a mistake, you need to fix only one place. Also, I personally do not recommend that you enclose everything in a function like bla7() because it makes things harder to debug. Your example could be rewritten like this:
The myGauss function;
A function w_ to calculate w1, w2, w10, w20, w100 and w200;
A function c_ to calculate c1, c2, c10, c20, c100 and c200;
A function normal_ to calculate normal1, normal2, normal10, normal20, normal100 and normal200;
Call all four functions as many times as needed with different inputs for different results.
If you do that, your could will look like this:
function x = myGauss(s,b_)
x = (1 / (s * sqrt(2 * %pi))) * exp(-0.5 * (b_/s)^2);
endfunction
function [w1_,w2_] = w_(t_,l_,n_,p_)
w1_ = zeros(t_,1);
w2_ = zeros(t_,1);
for I = 1 : t_
a = (grand(n_,1,"unf",0,p_));
x = l_ * cos(a);
y = l_ * sin(a);
z1 = zeros(n_,1);
z2 = zeros(n_,1);
for i = 2 : n_
z1(i) = z1(i-1) + x(i);
z2(i) = z2(i-1) + y(i);
end
w1_(I) = z1($);
w2_(I) = z2($);
end
endfunction
function [c1_,c2_] = c_(t_,k_,v_,w1_,x_)
c1_ = zeros(k_,1)
for r = 1 : t_
c = w1_(r);
m = -x_ + v_;
n = -x_;
for g = 1 : k_
if (c < m & c >= n) then
c1_(g) = c1_(g) + 1;
m = m + v_;
n = n + v_;
else
m = m + v_;
n = n + v_;
end
end
end
c2_ = zeros(k_,1);
c2_(1) = -x_ + (x_/k_);
for b = 2 : k_
c2_(b) = c2_(b-1) + v_;
end
endfunction
function [normal1_,normal2_,normal3_] = normal_(k_,bb_,bc_,v_,w1_)
y = stdev(w1_);
normal1_ = zeros(k_,1);
normal2_ = zeros(k_,1);
for wa = 1 : k_
bd_ = (bb_ + bc_) / 2;
gauss1 = myGauss(y,bb_);
gauss2 = myGauss(y,bc_);
gauss3 = myGauss(y,bd_);
gauss4 = ((bc_ - bb_) / 6) * (gauss1 + gauss2 + 4 * gauss3);
bb_ = bb_ + v_;
bc_ = bc_ + v_;
normal2_(wa,1) = gauss4;
end
normal3_ = normal2_ * 4000;
endfunction
t = 4000;
l = 0.067;
p = 2 * %pi;
n = 1000;
k = 70;
v = 12 / k;
x = 6;
bb = -x;
bc = -x + v;
[w1,w2] = w_(t,l,n,p);
[c1,c2] = c_(t,k,v,w1,x);
[normal1,normal2,normal3] = normal_(k,bb,bc,v,w1);
bar(c2,c1,1.0,'white');
plot(c2, normal3, 'r-');
n = 10000;
k = 100;
v = 24 / k;
x = 12;
bb = -x;
bc = -x + v;
[w10,w20] = w_(t,l,n,p);
[c10,c20] = c_(t,k,v,w10,x);
[normal10,normal20,normal30] = normal_(k,bb,bc,v,w10);
bar(c20,c10,1.0,'white');
plot(c20, normal30, 'b-');
n = 100;
k = 70;
v = 12 / k;
x = 6;
bb = -x;
bc = -x + v;
[w100,w200] = w_(t,l,n,p);
[c100,c200] = c_(t,k,v,w100,x);
[normal100,normal200,normal300] = normal_(k,bb,bc,v,w100);
bar(c200,c100,1.0,'white');
plot(c200, normal300, 'm-');
poly1.thickness=3;
xlabel(["x / um"]);
ylabel("molecules");
gcf().axes_size=[500,500]
a=gca();
a.zoom_box=[-12,12;0,600];
a.font_size=4;
a.labels_font_size=5;
a.x_label.font_size = 5;
a.y_label.font_size = 5;
ticks = a.x_ticks
ticks.labels =["-12";"-10";"-8";"-6";"-4";"-2";"0";"2";"4";"6";"8";"10";"12"]
ticks.locations = [-12;-10;-8;-6;-4;-2;0;2;4;6;8;10;12]
a.x_ticks = ticks
I created a function for a cumulative binomial distribution. It works well for extremely modest sample sizes, but I get a arithmetical overflow on larger samples.
The largest culprit is the n!. In excel 170! = 7.3E+306. 171! = #NUM!
Excel has an internal function that calculates binomial distribution, and it works with ns much, much larger than 170.
Is there something I can do to limit the magnitude of the #s generated?
EDIT: I played with this
SET #probout = 2*3*4*5*6*7*8*9*10*11*12
Worked fine
SET #probout = 2*3*4*5*6*7*8*9*10*11*12*13/10000
Resulted in overflow
Function below.
ALTER FUNCTION [dbo].[binomdist_cumulative]
(
#n int
,#k int
,#p float
)
RETURNS float
AS
BEGIN
-- Local Variable Declarations
-- ---------------------------
DECLARE #kfac float
,#nfac float
,#nkfac float
,#i float
,#f int
,#probout float
SET #i = 0
SET #f = 0
SET #nfac = 0
SET #kfac = 0
SET #nkfac = 0
SET #probout = 0
WHILE #i <= #k
BEGIN
--k!
SET #f = #i-1
SET #kfac = #i
IF #kfac > 0
BEGIN
WHILE #f > 0
BEGIN
SET #kfac = #kfac*#f
SET #f = #f -1
END
END
ELSE
BEGIN
SET #kfac = 1
END
--n!
SET #f = #n-1
SET #nfac = #n
IF #nfac > 0
BEGIN
WHILE #f > 0
BEGIN
SET #nfac = #nfac * #f
SET #f = #f -1
END
END
ELSE
BEGIN
SET #nfac = 1
END
--(n-k)!
SET #f = #n-#i-1
SET #nkfac = #n-#i
IF #nkfac > 0
BEGIN
WHILE #f > 0
BEGIN
SET #nkfac = #nkfac * #f
SET #f = #f -1
END
END
ELSE
BEGIN
SET #nkfac = 1
END
--Accumulate distribution
SET #probout = #probout + #nfac/(#kfac*#nkfac)*POWER(#p,#i)*POWER(1-#p,#n-#i)
SET #i = #i+1
END
RETURN #probout
END
Let me give you a hint.
If you calculate the full factorials, you are quickly going to get overflows. If you do an incremental calculation, the you won't.
For instance, instead of calculating (5 // 3) as (5*4*3*2*1) / ((3*2) * (3*2*1)), calculate it as: (5 / 3) * (4 / 2) * (3 / 3) * (2 / 2) * (1 / 1). . . oh, wait, you can see that the last three terms are all "1".
To be clear, you want to calculate the product of:
((n - i) / (n - k - i)
For i between 0 and k - 1. That is, you are dividing the product of k consecutive numbers ending in n with k consecutive numbers starting with 1.
You'll see that this incremental approach will forestall the issues with overflow.
I've been working on Project Euler questions as part of learning how to code in Lasso and am wondering if my solution can be improved upon. Here is what I've got below for question #1 in Lasso 8 code, and it returns the correct answer:
var ('total' = 0);
loop(1000-1);
loop_count % 3 == 0 || loop_count % 5 == 0 ? $total += loop_count;
/loop;
output($total);
My question: is there a better or faster way to code this? Thanks!
Actually Chris it looks like my L9 code answer was almost exactly the same. However what I had to do to time is was wrap it in a loop to time it 1000 times.
Lasso 9 can do Microseconds, whereas versions prior can only time in milliseconds.
Below are 3 ways - the first is yours, then my two versions.
define br => '<br>'
local(start_time = micros)
loop(1000)=>{
var ('total' = 0);
loop(1000-1);
loop_count % 3 == 0 || loop_count % 5 == 0 ? $total += loop_count;
/loop;
$total;
}
'Avg (L8 code in 9): '+(micros - #start_time)/1000+' micros'
br
br
local(start_time = micros)
loop(1000)=>{
local(sum = 0)
loop(999)=>{ loop_count % 3 == 0 || loop_count % 5 == 0 ? #sum += loop_count }
#sum
}
'Avg (incremental improvement): '+(micros - #start_time)/1000+' micros'
br
br
local(start_time = micros)
loop(1000)=>{
local(sum = 0)
loop(999)=>{ not (loop_count % 3) || not(loop_count % 5) ? #sum += loop_count }
#sum
}
'Avg using boolean not: '+(micros - #start_time)/1000+' micros'
The output is:
Avg (L8 code in 9): 637 micros
Avg (incremental improvement): 595 micros
Avg using boolean not: 596 micros
Note that I didn't use "output" as it's redundant in many situations in 8 and completely redundant 9 :)
There is a fun story about how Gauss once summed numbers, which involves a strategy which can help to avoid the loop.
local('p' = 3);
local('q' = 5);
local('n' = 1000);
local('x' = integer);
local('before');
local('after');
#before = micros
loop(1000) => {
/* In the tradition of Gauss */
local('n2' = #n - 1)
local('pq' = #p * #q)
local('p2' = #n2 / #p)
local('q2' = #n2 / #q)
local('pq2' = #n2 / #pq)
local('p3' = (#p2 + 1) * (#p2 / 2) + (#p2 % 2 ? #p2 / 2 + 1 | 0))
local('q3' = (#q2 + 1) * (#q2 / 2) + (#q2 % 2 ? #q2 / 2 + 1 | 0))
local('pq3' = (#pq2 + 1) * (#pq2 / 2) + (#pq2 % 2 ? #pq2 / 2 + 1 | 0))
#x = #p * #p3 + #q * #q3 - #pq * #pq3
}
#after = micros
'Answer: ' + #x + '<br/>\n'
'Average time: ' + ((#after - #before) / 1000) + '<br/>\n'
/* Different numbers */
#p = 7
#q = 11
#before = micros
loop(1000) => {
/* In the tradition of Gauss */
local('n2' = #n - 1)
local('pq' = #p * #q)
local('p2' = #n2 / #p)
local('q2' = #n2 / #q)
local('pq2' = #n2 / #pq)
local('p3' = (#p2 + 1) * (#p2 / 2) + (#p2 % 2 ? #p2 / 2 + 1 | 0))
local('q3' = (#q2 + 1) * (#q2 / 2) + (#q2 % 2 ? #q2 / 2 + 1 | 0))
local('pq3' = (#pq2 + 1) * (#pq2 / 2) + (#pq2 % 2 ? #pq2 / 2 + 1 | 0))
#x = #p * #p3 + #q * #q3 - #pq * #pq3
}
#after = micros
'Answer: ' + #x + '<br/>\n'
'Average time: ' + ((#after - #before) / 1000) + '<br/>\n'
The output is:
Answer: 233168<br/>
Average time: 3<br/>
Answer: 110110<br/>
Average time: 2<br/>
Although the first time I ran it, that first average time was 18 instead of 3. Maybe Lasso is doing something smart for subsequent runs, or maybe it was just bad luck.
n = input number
x = (n-1)/3 = count of 3 divisible numbers.*
sum3 = (3*x*(x+1)) / 2 = sum of those numbers.**
y = (n-1)/5 = count of 5 divisible numbers.
sum5 = (5*y*(y+1)) / 2 = sum of those numbers.
half_Ans = sum3 + sum5
but 15, 30, 45... count twice (in both sum3 & sum5).
so remove it one time, so only they count once.
z = (n-1)/15 = count of 15 divisible numbers.
sum15 = (15*z*(z+1)) / 2 = sum of those numbers.
Answer = half_Ans - sum15
* => (n-1)/3 gives count of 3 divisible numbers.
if n = 100 we need to count of (3, 6, 9, ..., 99)
3 is 1st, 6 is 2nd, .... so on 99 is 33rd
so total count of those number is gain by last number / 3
last number is near to our input n (specifically less than input n)
if n = 99 we must not count 99, because statement is "find the sum of all the multiples of 3 or 5 below n".
so w/o subtract 1 last unwanted number also count, if n is divisible by 3.
** => (3*x*(x+1)) / 2 gives sum of those numbers
if n = 100 sum id 3 + 6 + 9 + ... + 99
all component are multiple of 3.
so 3 + 6 + 9 + ... + 99 = 3*(1 + 2 + 3 + ... + 33)
sum of 1 to m is (m*(m+1)) / 2
so 3 + 6 + 9 + ... + 99 = (3*33*(33+1)) / 2
here m for 1 to m is last number or total number of that sequence
or length of sequence that's why we find count of divisible numbers.