So, I am creating a mathematical program, it has a couple of functions, one of the function is a decimal place holder, what this is:
the user is asked how many decimal places between 1 and 5. he would like his answers to be shown for the different mathematical function.
Let’s say the user chooses the answer to be to 4 decimal places, he then sets this up and he chooses the quadratic equation solver, he is asked for A,B,C values, once it is calculated the, let say he gives inputs of 1, -8 and 15, the answer is -3 and 5. However as he had chosen his answer to be to 4 decimal places the answer should be produced as -3.0000 and 5.0000
I can not get this to work, here is my code below, can someone help me solve this, been stuck on it for a couple of days now
I have tried to round the number by the Decimal place input from the user: not working
'Decimal place rule
Sub Accuracy()
Line1:
Dim DP
Console.WriteLine("Please Enter the Decimial Limit between 1-5: ")
DP = Double.Parse(Console.ReadLine())
If (DP > 5) Then
Console.WriteLine("Error, Decimial Limit is between 1 and 5, Please Try Again!")
GoTo Line1
Else
DP = DP
Console.Write("Decimial Limit has been Set Succuesfully to " & DP & " Decimal Places")
End If
End Sub
'Quadratic Equation function
Sub QuadraticFunction()
Dim a, b, c As Integer
Dim d, x1, x2 As Double
line1:
Console.WriteLine("Please Input a Non-Zero Number, A: ")
a = Console.ReadLine()
If (a = 0) Then
Console.WriteLine("Error, Number must not be 0, Try Again!")
GoTo line1
End If
Console.WriteLine("Please Input The Value of, B: ")
b = Console.ReadLine()
Console.Write("Please Input the Value of, C: ")
c = Console.ReadLine()
d = b * b - (4 * a * c)
If (d = 0) Then
Console.WriteLine("Both Roots Are Equal.")
x1 = -b / (2.0 * a)
x2 = x1
x1 = Math.Round(x1, DP)
x2 = Math.Round(x1, DP)
Console.WriteLine("First Root, (Root1) = {0}", x1)
Console.WriteLine("Second Root, (Root2) = {0}", x2)
ElseIf (d > 0) Then
Console.WriteLine("Both Roots are Real and Different")
x1 = (-b + Math.Sqrt(d)) / (2 * a)
x2 = (-b - Math.Sqrt(d)) / (2 * a)
x1 = (Math.Round(x1, DP))
x2 = (Math.Round(x2, DP))
Console.WriteLine("First Root, (Root1) = {0}", x1)
Console.WriteLine("Second Root, (Root2) = {0}", x2)
Else
Console.Write("Root are Imaginary " & "No Solution")
End If
End Sub
Related
Here I'll use the notation
It is possible to find the continued fraction of a number by computing it then applying the definition, but that requires at least O(n) bits of memory to find a0, a1 ... an, in practice it is a much worse. Using double floating point precision it is only possible to find a0, a1 ... a19.
An alternative is to use the fact that if a,b,c are rational numbers then there exist unique rationals p,q,r such that 1/(a+b*21/3+c*22/3) = x+y*21/3+z*22/3, namely
So if I represent x,y, and z to absolute precision using the boost rational lib I can obtain floor(x + y*21/3+z*22/3) accurately only using double precision for 21/3 and 22/3 because I only need it to be within 1/2 of the true value. Unfortunately the numerators and denominators of x,y, and z grow considerably fast, and if you use regular floats instead the errors pile up quickly.
This way I was able to compute a0, a1 ... a10000 in under an hour, but somehow mathematica can do that in 2 seconds. Here's my code for reference
#include <iostream>
#include <boost/multiprecision/cpp_int.hpp>
namespace mp = boost::multiprecision;
int main()
{
const double t_1 = 1.259921049894873164767210607278228350570251;
const double t_2 = 1.587401051968199474751705639272308260391493;
mp::cpp_rational p = 0;
mp::cpp_rational q = 1;
mp::cpp_rational r = 0;
for(unsigned int i = 1; i != 10001; ++i) {
double p_f = static_cast<double>(p);
double q_f = static_cast<double>(q);
double r_f = static_cast<double>(r);
uint64_t floor = p_f + t_1 * q_f + t_2 * r_f;
std::cout << floor << ", ";
p -= floor;
//std::cout << floor << " " << p << " " << q << " " << r << std::endl;
mp::cpp_rational den = (p * p * p + 2 * q * q * q +
4 * r * r * r - 6 * p * q * r);
mp::cpp_rational a = (p * p - 2 * q * r) / den;
mp::cpp_rational b = (2 * r * r - p * q) / den;
mp::cpp_rational c = (q * q - p * r) / den;
p = a;
q = b;
r = c;
}
return 0;
}
The Lagrange algorithm
The algorithm is described for example in Knuth's book The Art of Computer Programming, vol 2 (Ex 13 in section 4.5.3 Analysis of Euclid's Algorithm, p. 375 in 3rd edition).
Let f be a polynomial of integer coefficients whose only real root is an irrational number x0 > 1. Then the Lagrange algorithm calculates the consecutive quotients of the continued fraction of x0.
I implemented it in python
def cf(a, N=10):
"""
a : list - coefficients of the polynomial,
i.e. f(x) = a[0] + a[1]*x + ... + a[n]*x^n
N : number of quotients to output
"""
# Degree of the polynomial
n = len(a) - 1
# List of consecutive quotients
ans = []
def shift_poly():
"""
Replaces plynomial f(x) with f(x+1) (shifts its graph to the left).
"""
for k in range(n):
for j in range(n - 1, k - 1, -1):
a[j] += a[j+1]
for _ in range(N):
quotient = 1
shift_poly()
# While the root is >1 shift it left
while sum(a) < 0:
quotient += 1
shift_poly()
# Otherwise, we have the next quotient
ans.append(quotient)
# Replace polynomial f(x) with -x^n * f(1/x)
a.reverse()
a = [-x for x in a]
return ans
It takes about 1s on my computer to run cf([-2, 0, 0, 1], 10000). (The coefficients correspond to the polynomial x^3 - 2 whose only real root is 2^(1/3).) The output agrees with the one from Wolfram Alpha.
Caveat
The coefficients of the polynomials evaluated inside the function quickly become quite large integers. So this approach needs some bigint implementation in other languages (Pure python3 deals with it, but for example numpy doesn't.)
You might have more luck computing 2^(1/3) to high accuracy and then trying to derive the continued fraction from that, using interval arithmetic to determine if the accuracy is sufficient.
Here's my stab at this in Python, using Halley iteration to compute 2^(1/3) in fixed point. The dead code is an attempt to compute fixed-point reciprocals more efficiently than Python via Newton iteration -- no dice.
Timing from my machine is about thirty seconds, spent mostly trying to extract the continued fraction from the fixed point representation.
prec = 40000
a = 1 << (3 * prec + 1)
two_a = a << 1
x = 5 << (prec - 2)
while True:
x_cubed = x * x * x
two_x_cubed = x_cubed << 1
x_prime = x * (x_cubed + two_a) // (two_x_cubed + a)
if -1 <= x_prime - x <= 1: break
x = x_prime
cf = []
four_to_the_prec = 1 << (2 * prec)
for i in range(10000):
q = x >> prec
r = x - (q << prec)
cf.append(q)
if True:
x = four_to_the_prec // r
else:
x = 1 << (2 * prec - r.bit_length())
while True:
delta_x = (x * ((four_to_the_prec - r * x) >> prec)) >> prec
if not delta_x: break
x += delta_x
print(cf)
I want to select a random number from 0,1,2,3...n, however I want to make it that the chance of selecting k|0<k<n will be lower by multiplication of x from selecting k - 1 so x = (k - 1) / k. As bigger the number as smaller the chances to pick it up.
As an answer I want to see the implementation of the next method:
int pickANumber(n,x)
This is for a game that I am developing, I saw those questions as related but not exactly that same:
How to pick an item by its probability
C Function for picking from a list where each element has a distinct probabili
p1 + p2 + ... + pn = 1
p1 = p2 * x
p2 = p3 * x
...
p_n-1 = pn * x
Solving this gives you:
p1 + p2 + ... + pn = 1
(p2 * x) + (p3 * x) + ... + (pn * x) + pn = 1
((p3*x) * x) + ((p4*x) * x) + ... + ((p_n-1*x) * x) + pn = 1
....
pn* (x^(n-1) + x^(n-2) + ... +x^1 + x^0) = 1
pn*(1-x^n)/(1-x) = 1
pn = (1-x)/(1-x^n)
This gives you the probability you need to set to pn, and from it you can calculate the probabilities for all other p1,p2,...p_n-1
Now, you can use a "black box" RNG that chooses a number with a distribution, like those in the threads you mentioned.
A simple approach to do it is to set an auxillary array:
aux[i] = p1 + p2 + ... + pi
Now, draw a random number with uniform distribution between 0 to aux[n], and using binary search (aux array is sorted), get the first value, which matching value in aux is greater than the random uniform number you got
Original answer, for substraction (before question was editted):
For n items, you need to solve the equation:
p1 + p2 + ... + pn = 1
p1 = p2 + x
p2 = p3 + x
...
p_n-1 = pn + x
Solving this gives you:
p1 + p2 + ... + pn = 1
(p2 + x) + (p3 + x) + ... + (pn + x) + pn = 1
((p3+x) + x) + ((p4+x) + x) + ... + ((p_n-1+x) + x) + pn = 1
....
pn* ((n-1)x + (n-2)x + ... +x + 0) = 1
pn* x = n(n-1)/2
pn = n(n-1)/(2x)
This gives you the probability you need to set to pn, and from it you can calculate the probabilities for all other p1,p2,...p_n-1
Now, you can use a "black box" RNG that chooses a number with a distribution, like those in the threads you mentioned.
Be advised, this is not guaranteed you will have a solution such that 0<p_i<1 for all i, but you cannot guarantee one given from your requirements, and it is going to depend on values of n and x to fit.
Edit This answer was for the OPs original question, which was different in that each probability was supposed to be lower by a fixed amount than the previous one.
Well, let's see what the constraints say. You want to have P(k) = P(k - 1) - x. So we have:
P(0)
P(1) = P(0) - x
P(2) = P(0) - 2x
...
In addition, Sumk P(k) = 1. Summing, we get:
1 = (n + 1)P(0) -x * n / 2 (n + 1),
This gives you an easy constraint between x and P(0). Solve for one in terms of the other.
For this I would use the Mersenne Twister algorithm for a uniform distribution which Boost provides, then have a mapping function to map the results of that random distribution to the actual number select.
Here's a quick example of a potential implementation, although I left out the quadtratic equation implementation since it is well known:
int f_of_xib(int x, int i, int b)
{
return x * i * i / 2 + b * i;
}
int b_of_x(int i, int x)
{
return (r - ( r ) / 2 );
}
int pickANumber(mt19937 gen, int n, int x)
{
// First, determine the range r required where the probability equals i * x
// since probability of each increasing integer is x higher of occuring.
// Let f(i) = r and given f'(i) = x * i then r = ( x * i ^2 ) / 2 + b * i
// where b = ( r - ( x * i ^ 2 ) / 2 ) / i . Since r = x when i = 1 from problem
// definition, this reduces down to b = r - r / 2. therefore to find r_max simply
// plugin x to find b, then plugin n for i, x, and b to get r_max since r_max occurs
// when n == i.
// Find b when
int b = b_of_x(x);
int r_max = f_of_xib(x, n, b);
boost::uniform_int<> range(0, r_max);
boost::variate_generator<boost::mt19937&, boost::uniform_int<> > next(gen, range);
// Now to map random number to desired number, just find the positive value for i
// when r is the return random number which boils down to finding the non-zero root
// when 0 = ( x * i ^ 2 ) / 2 + b * i - r
int random_number = next();
return quadtratic_equation_for_positive_value(1, b, r);
}
int main(int argc, char** argv)
{
mt19937 gen;
gen.seed(time(0));
pickANumber(gen, 10, 1);
system("pause");
}
I have a formula of a sequence of double numbers k = a + d * n, where a and d are constant double values, n is an integer number, k >= 0, a >= 0. For example:
..., 300, 301.6, 303.2, 304.8, 306.4, ...
I want to round a given number c to a nearest value from this sequence which is lower than c.
Currently I use something like this:
double someFunc(double c) {
static double a = 1;
static double d = 2;
int n = 0;
double a1 = a;
if (c >= a) {
while (a1 < c) {
a1 += d;
}
a1 -= d;
} else {
while (a1 > c) {
a1 -= d;
}
}
return a1;
}
Is it possible to do the same without these awful cycles? I ask because the following situation may appear:
abs(a - c) >> abs(d) (the first number is much more then the second one and so a lot of iterations possible)
My question is similar to the following one. But in my case I also have a a variable which has influence on the final result. It means that a sequence may haven't number 0.
Suppose c is a number in your sequence. Then you have n = (c - a) / d.
Since you want an integer <= c, then take n = floor((c - a) / d).
Then you can round c to: a + d * floor((c - a) / d)
Suppose k = 3 + 5 * n and you round c=21.
And 3 + 5 * floor((21 - 3) / 5) = 3 + 5 * 3 = 18
Below I've posted the code to a non-working "divide and conquer" multiplication method in ruby(with debug prints). I cannot tell if its broken code, or a quirk in Ruby like how the L-shift(<<) operator doesn't push bits into the bit-bucket; this is unexpected compared to similar operations in C++.
Is it broken code (doesn't match the original algorithm) or unexpected behavior?
Pseudo code for original algorithm
def multiply(x,y,n, level)
#print "Level #{level}\n"
if n == 1
#print "\tx[#{x.to_s(2)}][#{y.to_s(2)}]\n\n"
return x*y
end
mask = 2**n - 2**(n/2)
xl = x >> (n / 2)
xr = x & ~mask
yl = y >> (n / 2)
yr = y & ~mask
print " #{n} | x = #{x.to_s(2)} = L[#{xl.to_s(2)}][#{xr.to_s(2)}]R \n"
print " #{n} | y = #{y.to_s(2)} = L[#{yl.to_s(2)}][#{yr.to_s(2)}]R \n"
#print "\t[#{xl.to_s(2)}][#{yr.to_s(2)}]\n"
#print "\t[#{xr.to_s(2)}][#{yr.to_s(2)}]\n"
#print "\t([#{xl.to_s(2)}]+[#{xr.to_s(2)}])([#{yl.to_s(2)}]+[#{yr.to_s(2)}])\n\n"
p1 = multiply( xl, yl, n/2, level+1)
p2 = multiply( xr, yr, n/2, level+1)
p3 = multiply( xl+xr, yl+yr, n/2, level+1)
return p1 * 2**n + (p3 - p1 - p2) * 2**(n/2) + p2
end
x = 21
y = 22
print "x = #{x} = #{x.to_s(2)}\n"
print "y = #{y} = #{y.to_s(2)}\n"
print "\nDC_multiply\t#{x}*#{y} = #{multiply(x,y,8, 1)} \nregular\t#{x}*#{y} = #{x*y}\n\n "
I am not familiar with the divide and conquer algorithm but i don't think it contains parts you can't do in Ruby.
Here is a quick attempt:
def multiplb(a,b)
#Break recursion when a or b has one digit
if a < 10 || b < 10
a * b
else
#Max number of digits of a and b
n = [a.to_s.length, b.to_s.length].max
# Steps to split numbers to high and low digits sub-numbers
# (1) to_s.split('') => Converting digits to string then arrays to ease splitting numbers digits
# (2) each_slice => Splitting both numbers to high(left) and low(right) digits groups
# (3) to_a , map, join, to_i => Simply returning digits to numbers
al, ar = a.to_s.split('').each_slice(n/2).to_a.map(&:join).map(&:to_i)
bl, br = b.to_s.split('').each_slice(n/2).to_a.map(&:join).map(&:to_i)
#Recursion
p1 = multiplb(al, bl)
p2 = multiplb(al + ar, bl + br)
p3 = multiplb(ar, br)
p1 * (10**n) + (p2 - p1 - p3) * (10**(n/2)) + p3
end
end
#Test
puts multiplb(1980, 2315)
# => 4583700 yeah that's correct :)
Here are some references to further explain part of the code:
Finding max of numbers => How do you find a min / max with Ruby?
Spliting an array to half => Splitting an array into equal parts in ruby
Turning a fixnum into array => Turning long fixed number to array Ruby
Hope it hepls !
There are two random functions f1(),f2().
f1() returns 1 with probability p1, and 0 with probability 1-p1.
f2() returns 1 with probability p2, and 0 with probability 1-p2.
I want to implement a new function f3() which returns 1 with probability p3(a given probability), and returns 0 with probability 1-p3. In the implemetion of function f3(), we can use function f1() and f2(), but you can't use any other random function.
If p3=0.5, an example of implemention:
int f3()
{
do
{
int a = f1();
int b = f1();
if (a==b) continue;
// when reachs here
// a==1 with probability p1(1-p1)
// b==1 with probability (1-p1)p1
if (a==1) return 1;//now returns 1 with probability 0.5
if (b==1) return 0;
}while(1)
}
This implemention of f3() will give a random function returns 1 with probability 0.5, and 0 with probability 0.5. But how to implement the f3() with p3=0.4? I have no idea.
I wonder, is that task possible? And how to implement f3()?
Thanks in advance.
p1 = 0.77 -- arbitrary value between 0 and 1
function f1()
if math.random() < p1 then
return 1
else
return 0
end
end
-- f1() is enough. We don't need f2()
p3 = 0.4 -- arbitrary value between 0 and 1
--------------------------
function f3()
left = 0
rigth = 1
repeat
middle = left + (right - left) * p1
if f1() == 1 then
right = middle
else
left = middle
end
if right < p3 then -- completely below
return 1
elseif left >= p3 then -- completely above
return 0
end
until false -- loop forever
end
This can be solved if p3 is a rational number.
We should use conditional probabilities for this.
For example, if you want to make this for p3=0.4, the method is the following:
Calculate the fractional form of p3. In our case it is p3=0.4=2/5.
Now generate as many random variables from the same distribution (let's say, from f1, we won't use f2 anyway) as the denominator, call them X1, X2, X3, X4, X5.
We should regenerate all these random X variables until their sum equals the numerator in the fractional form of p3.
Once this is achieved then we just return X1 (or any other Xn, where n was chosen independently of the values of the X variables). Since there are 2 1s among the 5 X variables (because their sum equals the numerator), the probability of X1 being 1 is exactly p3.
For irrational p3, the problem cannot be solved by using only f1. I'm not sure now, but I think, it can be solved for p3 of the form p1*q+p2*(1-q), where q is rational with a similar method, generating the appropriate amount of Xs with distribution f1 and Ys with distribution f2, until they have a specific predefined sum, and returning one of them. This still needs to be detailed.
First to say, that's a nice problem to tweak one's brain. I managed to solve the problem for p3 = 0.4, for what you just asked for! And I think, generalisation of such problem, is not so trivial. :D
Here is how, you can solve it for p3 = 0.4:
The intuition comes from your example. If we generate a number from f1() five times in an iteration, (see the code bellow), we can have 32 types of results like bellow:
1: 00000
2: 00001
3: 00010
4: 00011
.....
.....
32: 11111
Among these, there are 10 such results with exactly two 1's in it! After identifying this, the problem becomes simple. Just return 1 for any of the 4 combinations and return 0 for 6 others! (as probability 0.4 means getting 1, 4 times out of 10). You can do that like bellow:
int f3()
{
do{
int a[5];
int numberOfOneInA = 0;
for(int i = 0; i < 5; i++){
a[i] = f1();
if(a[i] == 1){
numberOfOneInA++;
}
}
if (numberOfOneInA != 2) continue;
else return a[0]; //out of 10 times, 4 times a[0] is 1!
}while(1)
}
Waiting to see a generalised solution.
Cheers!
Here is an idea that will work when p3 is of a form a/2^n (a rational number with a denominator that is a power of 2).
Generate n random numbers with probability distribution of 0.5:
x1, x2, ..., xn
Interpret this as a binary number in the range 0...2^n-1; each number in this range has equal probability. If this number is less than a, return 1, else return 0.
Now, since this question is in a context of computer science, it seems reasonable to assume that p3 is in a form of a/2^n (this a common representation of numbers in computers).
I implement the idea of anatolyg and Egor:
inline double random(void)
{
return static_cast<double>(rand()) / static_cast<double>(RAND_MAX);
}
const double p1 = 0.8;
int rand_P1(void)
{
return random() < p1;
}
int rand_P2(void)//return 0 with 0.5
{
int x, y; while (1)
{
mystep++;
x = rand_P1(); y = rand_P1();
if (x ^ y) return x;
}
}
double p3 = random();
int rand_P3(void)//anatolyg's idea
{
double tp = p3; int bit, x;
while (1)
{
if (tp * 2 >= 1) {bit = 1; tp = tp * 2 - 1;}
else {bit = 0; tp = tp * 2;}
x = rand_P2();
if (bit ^ x) return bit;
}
}
int rand2_P3(void)//Egor's idea
{
double left = 0, right = 1, mid;
while (1)
{
dashenstep++;
mid = left + (right - left) * p1;
int x = rand_P1();
if (x) right = mid; else left = mid;
if (right < p3) return 1;
if (left > p3) return 0;
}
}
With massive math computings, I get, assuming P3 is uniformly distributed in [0,1), then the expectation of Egor is (1-p1^2-(1-p1)^2)^(-1). And anatolyg is 2(1-p1^2-(1-p1)^2)^(-1).
Speaking Algorithmically , Yes It is possible to do that task done .
Even Programmatically , It is possible , but a complex problem .
Lets take an example .
Let
F1(1) = .5 which means F1(0) =.5
F2(2) = .8 which means F1(0) =.2
Let Suppose You need a F3, such that F3(1)= .128
Lets try Decomposing it .
.128
= (2^7)*(10^-3) // decompose this into know values
= (8/10)*(8/10)*(2/10)
= F2(1)&F2(1)*(20/100) // as no Fi(1)==2/10
= F2(1)&F2(1)*(5/10)*(4/10)
= F2(1)&F2(1)&F1(1)*(40/100)
= F2(1)&F2(1)&F1(1)*(8/10)*(5/10)
= F2(1)&F2(1)&F1(1)&F2(1)&F1(1)
So F3(1)=.128 if we define F3()=F2()&F2()&F2()&F1()&F1()
Similarly if you want F4(1)=.9 ,
You give it as F4(0)=F1(0) | F2(0) =F1(0)F2(0)=.5.2 =.1 ,which mean F4(1)=1-0.1=0.9
Which means F4 is zero only when both are zero which happens .
So making use this ( & , | and , not(!) , xor(^) if you want ) operations with a combinational use of f1,f2 will surely give you the F3 which is made purely out of f1,f2,
Which may be NP hard problem to find the combination which gives you the exact probability.
So, Finally the answer to your question , whether it is possible or not ? is YES and this is one way of doing it, may be many hacks can be made into it this to optimize this, which gives you any optimal way .