If
I = (V+13) % 26
Then what is V in terms of 'I'. Basically how can you take the Mod operator on the other side of equation ?
I = (V+13) %26
This means V+13 = k*26 + I, and k=Z, I=[0,1,...,26-1]. (1)
so
V = k*26 + I -13
now, because I is remainder from division by 26, I=[0,1,...,26-1].
This means:
I%26=I, I divided by 26 is just I.
so:
I = (V+13) %26
I%26 = (V+13) %26
this is not the same as I%26 = (V+13)
because I%26 = (V+13) implies V+13 = [0,1,...,26-1] what contradicts (1), (V+13 might be greater than 26-1).
Corollary:
V = (I%26) - 13 is not correct
V = (I%26) - 13 + k*26, k=Z is correct
Related
How can I write a Vigenère encryption in Qbasic without the use of arrays?
I understand the math to encrypt a message:
Ca = Ma + Kb (mod 26)
And to decrypt a message:
Ma = Ca – Kb (mod 26).
I'm struggling with the syntax as I haven't found much information online.
You can solve this easily without using any arrays.
Below is my all-(Q)BASIC solution.
The MID$ function extracts one character from a string, and the ASC function converts the character into its ASCII code. Subtracting 65 produces a number in the range [0,25]. The encrypted number is turned back into a character using the CHR$ function. Hereafter the MID$ statement is used to put the encrypted character back in the string.
Because of the different lengths between the message and the encryption key, a separate iteration variable (j%) is needed to repeatedly walk through the key string.
msg$ = "ENCRYPTION"
PRINT msg$
key$ = "CLINTON"
k% = LEN(key$)
j% = 1
FOR i% = 1 TO LEN(msg$)
a% = (ASC(MID$(msg$, i%, 1)) - 65) + (ASC(MID$(key$, j%, 1)) -65)
MID$(msg$, i%) = CHR$(65 + a% + 26 * (a% > 25))
j% = j% + 1 + k% * (j% = k%)
NEXT i%
PRINT msg$
The above snippet could do without one - 65 and one + 65, but I've left these in for clarity.
The decryption process is quite similar. 3 little changes are all that it takes:
j% = 1
FOR i% = 1 TO LEN(msg$)
a% = (ASC(MID$(msg$, i%, 1)) - 65) - (ASC(MID$(key$, j%, 1)) -65)
MID$(msg$, i%) = CHR$(65 + a% - 26 * (a% < 0))
j% = j% + 1 + k% * (j% = k%)
NEXT i%
PRINT msg$
Running both snippets in a row produces:
ENCRYPTION
GYKERDGKZV
ENCRYPTION
What about a version of the code that can deal with spaces, punctuation marks, and accented characters?
The code is very similar and even a bit simpler:
msg$ = "This is any text that needs encrypting. So sayeth Sep Roland!"
PRINT msg$
key$ = "Blaise de Vigenère"
k% = LEN(key$)
j% = 1
FOR i% = 1 TO LEN(msg$)
a% = ASC(MID$(msg$, i%, 1)) + ASC(MID$(key$, j%, 1))
MID$(msg$, i%) = CHR$(a% + 256 * (a% > 255))
j% = j% + 1 + k% * (j% = k%)
NEXT i%
PRINT msg$
j% = 1
FOR i% = 1 TO LEN(msg$)
a% = ASC(MID$(msg$, i%, 1)) - ASC(MID$(key$, j%, 1))
MID$(msg$, i%) = CHR$(a% - 256 * (a% < 0))
j% = j% + 1 + k% * (j% = k%)
NEXT i%
PRINT msg$
I will not reproduce any output here because that would be a real pita...
Are these strange embedded conditions correct?
(a% + 26 * (a% > 25))
Consider the equivalent simple code:
IF a% > 25 THEN
a% = a% - 26
ENDIF
If the a% variable is greater than 25, we need to subtract 26.
Nonetheless the (a% + 26 * (a% > 25)) form uses addition.
This so happens because a TRUE condition evaluates to -1.
If a% > 25 is TRUE we get (a% + 26 * -1) -> a% - 26
If a% > 25 is FALSE we get (a% + 26 * 0) -> a%
You can simply get the ASCII value of the char as a number and then subtract the character value of A. You would get a number in the range [0, 26). Then you'd perform encryption / decryption as you've stated. To get back a valid character value then reverse and add the value of A. This works because the letters of the English alphabet (the ABC) are listed in order in ASCII.
To get the ciphertext or plaintext simply iterate over all the characters in the string (possibly after checking that it doesn't contain any other characters) and append the encrypted / decrypted character to a new string, and finally return that. Viola, no arrays, just strings, characters and numerical values.
That's all folks.
I have a simple question. How do I use the command qpsolve from Scilab if I only want to use the lower bounds and upper bounds limit?
ci <= x <= cs
The command can be used as this:
[x [,iact [,iter [,f]]]] = qpsolve(Q,p,C,b,ci,cs,me)
But I want to use it like this:
x = qpsolve(Q,p,[],[],ci,cs,[])
Only ci and cs should explain the limits for vector x. Unfortunately, the command cannot take empty []. Should I take [] as a row vector of ones or zeros?
https://help.scilab.org/docs/6.0.1/en_US/qpsolve.html
In Scilab 5.5.1 , [] works for C and b but not for me. so C = [];b = [];me = 0; should work.
Why
qpsolve is an interface for qp_solve :
function [x, iact, iter, f]=qpsolve(Q,p,C,b,ci,cs,me)
rhs = argn(2);
if rhs <> 7
error(msprintf(gettext("%s: Wrong number of input argument(s): %d expected.\n"), "qpsolve", 7));
end
C(me+1:$, :) = -C(me+1:$, :);
b(me+1:$) = -b(me+1:$);
// replace boundary contraints by linear constraints
Cb = []; bb = [];
if ci <> [] then
Cb = [Cb; speye(Q)]
bb = [bb; ci]
end
if cs <> [] then
Cb = [Cb; -speye(Q)]
bb = [bb; -cs]
end
C = [C; Cb]; b = [b; bb]
[x, iact, iter, f] = qp_solve(Q, -p, C', b, me)
endfunction
It transform every bound constraints into linear constraints. To begin, it swap the sign of the inequality constraints. To do that, it must know me, ie it must be an integer. Since C and b are empty matrices, is value doesn't matter.
Bonus:
if Q is inversible, you could skip the qpsolve macro and write
x = -Q\p
x(x<ci) = ci(x<ci)
x(x>cs) = cs(x>cs)
I'm trying solve the question of "Can x * (1/x) ever not be 1 when x is a random floating point number between [1,2]" and I am trying to generate random floating numbers in Julia to test the hypotheses. I've tried doing
BigFloat(rand(1,2)), as well as Float64(1,2)
to no avail. Any help is appreciated
The answers is yes
for i=1:100
x = rand() + 1.0;
xi = 1.0/x
y = x * xi
if y != 1.0
println("case ", i, " x*(1/x) != 1 for x=", x, " diff= ", y - 1.0)
end
end
case 18 x*(1/x) != 1 for x=1.3193289816663771 diff= -1.1102230246251565e-16
case 26 x*(1/x) != 1 for x=1.9692333690500858 diff= -1.1102230246251565e-16
case 42 x*(1/x) != 1 for x=1.8927527081187694 diff= -1.1102230246251565e-16
...
Aware this is due to the limited precision of floats (doubles).
This is not true in a mathematical sense.
My first Fortran lesson is to plot the probability density function of the radial Sturmian functions. In case you are interested, the radial Sturmian functions are used to graph the momentum space eigenfunctions for the hydrogen atom.
In order to produce these radial functions, one needs to first produce some polynomials called the Gegenbauer polynomials, denoted
Cba(x),
where a and b should be stacked atop each other. One needs these polynomials because the Sturmians (let's call them R_n,l) are defined like so,
R_n,l(p) = N pl⁄(p2 + k2)l+2 Cn - l - 1l + 1(p2 - k2⁄p2 + k2),
where N is a normalisation constant, p is the momentum, n is the principle quantum number, l is the angular momentum and k is a constant. The normalisation constant is there so that when I come to square this function, it will produce a probability distribution for the momentum of the electron in a hydrogen atom.
Gegenbauer polynomials are generated using the following recurrence relation:
Cnl(x) = 1⁄n[2(l + n - 1) x Cn - 1l(x) - (2l + n - 2)Cn - 2l(x)],
with C0l(x) = 1 and C1l(x) = 2lx, as you may have noticed, l is fixed but n is not. At the start of my program, I will specify both l and n and work out the Gegenbauer polynomial I need for the radial function I wish to plot.
The problems I am having with my code at the moment are all in my subroutine for working out the value of the Gegenbauer polynomial Cn-l-1l+1(p2 - k2⁄p2 + k2) for incremental values of p between 0 and 3. I keep getting the error
Unclassified statement at (1)
but I cannot see what the issue is.
program Radial_Plot
implicit none
real, parameter :: pi = 4*atan(1.0)
integer, parameter :: top = 1000, l = 50, n = 100
real, dimension(1:top) :: x, y
real increment
real :: a=0.0, b = 2.5, k = 0.3
integer :: i
real, dimension(1:top) :: C
increment = (b-a)/(real(top)-1)
x(1) = 0.0
do i = 2, top
x(i) = x(i-1) + increment
end do
Call Gegenbauer(top, n, l, k, C)
y = x*C
! y is the function that I shall be plotting between values a and b.
end program Radial_Plot
Subroutine Gegenbauer(top1, n1, l1, k1, CSub)
! This subroutine is my attempt to calculate the Gegenbauer polynomials evaluated at a certain number of values between c and d.
implicit none
integer :: top1, i, j, n1, l1
real :: k1, increment1, c, d
real, dimension(1:top1) :: x1
real, dimension(1:n1 - l1, 1:top1) :: C1
real, dimension(1:n1 - l1) :: CSub
c = 0.0
d = 3.0
k1 = 0.3
n1 = 50
l1 = 25
top1 = 1000
increment1 = (d - c)/(real(top1) - 1)
x1(1) = 0.0
do i = 2, top1
x1(i) = x1(i-1) + increment1
end do
do j = 1, top1
C1(1,j) = 1
C1(2,j) = 2(l1 + 1)(x1(i)^2 - k1^2)/(x1(i)^2 + k1^2)
! All the errors occurring here are all due to, and I quote, 'Unclassifiable statement at (1)', I can't see what the heck I have done wrong.
do i = 3, n1 - l1
C1(i,j) = 2(((l1 + 1)/n1) + 1)(x1(i)^2 - k1^2)/(x1(i)^2 + k1^2)C1(i,j-1) - ((2(l1+1)/n1) + 1)C1(i,j-2)
end do
CSub(j) = Cn(n1 - l1,j)^2
end do
return
end Subroutine Gegenbauer
As francesalus correctly pointed out, the problem is because you use ^ instead of ** for exponentiation. Additionally, you do not put * between the terms you are multiplying.
C1(1,j) = 1
C1(2,j) = 2*(l1 + 1)*(x1(i)**2 - k1**2)/(x1(i)**2 + k1**2)
do i = 3, n1 - l1
C1(i,j) = 2 * (((l1 + 1)/n1) + 1) * (x1(i)**2 - k1**2) / &
(x1(i)**2 + k1**2)*C1(i,j-1) - ((2(l1+1)/n1) + 1) * &
C1(i,j-2)
end do
CSub(j) = Cn(n1 - l1,j)**2
Since you are beginning I have some advice. Learn to put all subroutines and functions to modules (unless they are internal). There is no reason for the return statement at the and of the subroutine, similarly as a stop statement isn't necessary at the and of the program.
I have this recurrence formula:
P(n) = ( P(n-1) + 2^(n/2) ) % (X)
s.t. P(1) = 2;
where n/2 is computer integer division i.e. floor of x/2
Since i am taking mod X, this relation should repeat at least with in X outputs.
but it can start repeating before that.
How to find this value?
It needn't repeat within x terms, consider x = 3:
P(1) = 2
P(2) = (P(1) + 2^(2/2)) % 3 = 4 % 3 = 1
P(3) = (P(2) + 2^(3/2)) % 3 = (1 + 2) % 3 = 0
P(4) = (P(3) + 2^(4/2)) % 3 = 4 % 3 = 1
P(5) = (P(4) + 2^(5/2)) % 3 = (1 + 4) % 3 = 2
P(6) = (P(5) + 2^(6/2)) % 3 = (2 + 8) % 3 = 1
P(7) = (P(6) + 2^(7/2)) % 3 = (1 + 8) % 3 = 0
P(8) = (P(7) + 2^(8/2)) % 3 = 16 % 3 = 1
P(9) = (P(8) + 2^(9/2)) % 3 = (1 + 16) % 3 = 2
P(10) = (P(9) + 2^(10/2)) % 3 = (2 + 32) % 3 = 1
P(11) = (P(10) + 2^(11/2)) % 3 = (1 + 32) % 3 = 0
P(12) = (P(11) + 2^(12/2)) % 3 = (0 + 64) % 3 = 1
and you see that the period is 4.
Generally (suppose X is odd, it's a bit more involved for even X), let k be the period of 2 modulo X, i.e. k > 0, 2^k % X = 1, and k is minimal with these properties (see below).
Consider all arithmetic modulo X. Then
n
P(n) = 2 + ∑ 2^(j/2)
j=2
It is easier to see when we separately consider odd and even n:
m m
P(2*m+1) = 2 + 2 * ∑ 2^i = 2 * ∑ 2^i = 2*(2^(m+1) - 1) = 2^((n+2)/2) + 2^((n+1)/2) - 2
i=1 i=0
since each 2^j appears twice, for j = 2*i and j = 2*i+1. For even n = 2*m, there's one summand 2^m missing, so
P(2*m) = 2^(m+1) + 2^m - 2 = 2^((n+2)/2) + 2^((n+1)/2) - 2
and we see that the length of the period is 2*k, since the changing parts 2^((n+1)/2) and 2^((n+2)/2) have that period. The period immediately begins, there is no pre-period part (there can be a pre-period for even X).
Now k <= φ(X) by Euler's generalisation of Fermat's theorem, so the period is at most 2 * φ(X).
(φ is Euler's totient function, i.e. φ(n) is the number of integers 1 <= k <= n with gcd(n,k) = 1.)
What makes it possible that the period is longer than X is that P(n+1) is not completely determined by P(n), the value of n also plays a role in determining P(n+1), in this case the dependence is simple, each power of 2 being used twice in succession doubles the period of the pure powers of 2.
Consider the sequence a[k] = (2^k) % X for odd X > 1. It has the simple recurrence
a[0] = 1
a[k+1] = (2 * a[k]) % X
so each value completely determines the next, thus the entire following part of the sequence. (Since X is assumed odd, it also determines the previous value [if k > 0] and thus the entire previous part of the sequence. With H = (X+1)/2, we have a[k-1] = (H * a[k]) % X.)
Hence if the sequence assumes one value twice (and since there are only X possible values, that must happen within the first X+1 values), at indices i and j = i+p > i, say, the sequence repeats and we have a[k+p] = a[k] for all k >= i. For odd X, we can go back in the sequence, therefore a[k+p] = a[k] also holds for 0 <= k < i. Thus the first value that occurs twice in the sequence is a[0] = 1.
Let p be the smallest positive integer with a[p] = 1. Then p is the length of the smallest period of the sequence a, and a[k] = 1 if and only if k is a multiple of p, thus the set of periods of a is the set of multiples of p. Euler's theorem says that a[φ(X)] = 1, from that we can conclude that p is a divisor of φ(X), in particular p <= φ(X) < X.
Now back to the original sequence.
P(n) = 2 + a[1] + a[1] + a[2] + a[2] + ... + a[n/2]
= a[0] + a[0] + a[1] + a[1] + a[2] + a[2] + ... + a[n/2]
Since each a[k] is used twice in succession, it is natural to examine the subsequences for even and odd indices separately,
E[m] = P(2*m)
O[m] = P(2*m+1)
then the transition from one value to the next is more regular. For the even indices we find
E[m+1] = E[m] + a[m] + a[m+1] = E[m] + 3*a[m]
and for the odd indices
O[m+1] = O[m] + a[m+1] + a[m+1] = O[m] + 2*a[m+1]
Now if we ignore the modulus for the moment, both E and O are geometric sums, so there's an easy closed formula for the terms. They have been given above (in slightly different form),
E[m] = 3 * 2^m - 2 = 3 * a[m] - 2
O[m] = 2 * 2^(m+1) - 2 = 2 * a[m+1] - 2 = a[m+2] - 2
So we see that O has the same (minimal) period as a, namely p, and E also has that period. Unless maybe if X is divisible by 3, that is also the minimal (positive) period of E (if X is divisible by 3, the minimal positive period of E could be a proper divisor of p, for X = 3 e.g., E is constant).
Thus we see that 2*p is a period of the sequence P obtained by interlacing E and O.
It remains to be seen that 2*p is the minimal positive period of P. Let m be the minimal positive period. Then m is a divisor of 2*p.
Suppose m were odd, m = 2*j+1. Then
P(1) = P(m+1) = P(2*m+1)
P(2) = P(m+2) = P(2*m+2)
and consequently
P(2) - P(1) = P(m+2) - P(m+1) = P(2*m+2) - P(2*m+1)
But P(2) - P(1) = a[1] and
P(m+2) - P(m+1) = a[(m+2)/2] = a[j+1]
P(2*m+2) - P(2*m+1) = a[(2*m+2)/2] = a[m+1] = a[2*j+2]
So we must have a[1] = a[j+1], hence j is a period of a, and a[j+1] = a[2*j+2], hence j+1 is a period of a too. But that means that 1 is a period of a, which implies X = 1, a contradiction.
Therefore m is even, m = 2*j. But then j is a period of O (and of E), thus a multiple of p. On the other hand, m <= 2*p implies j <= p, and the only (positive) multiple of p satisfying that inequality is p itself, hence j = p, m = 2*p.