solve the the following recuurence relation when k= √n - search

T(n) = T(n/k) + (k − 1)
Suppose we set 𝑘 =. √n how to Find asymptotic running of T(n). any ideas or help is appreciated ..thanks

Related

The divide and conquer problem algorithms

a - O(n^(log_3 4))
b - Theta(n log n)
c - Theta(n^2).
d - O(n)
e - Theta(n^(log_4 3))
Essentially your problem is a recurrence relationship like this;
T(n) = 5*T(n/3) + Theta(n^2)
You can use what is called the Master Theorem
to get an answer for this.
Your parameters are: a:=3 b:=5 and f(n):=n^2. I'm sure from here you can solve your question and get your answer.

How to solve this optimization problem with cvxopt

I have a non-linear optimization problem which, in Mathematica, could be solved as:
FindMaximum[{(81 x + 19)^0.4 + (80 (1 - x) + 20)^0.6, 0 <= x <= 1}, x‬‬]
However, now I am on a computer without Mathematica and I would like to solve a similar problem in Python, using the CVXOPT module. I looked at the examples and found linear programs, quadratic programs, and other kinds of programs, but could not find this simple program.
Can I solve such a program with CVXOPT?
I did not find a solution with cvxopt, but I found a much better alternative - cvxpy:
import cvxpy as cp
x = cp.Variable()
prob = cp.Problem(
cp.Maximize((81*x + 19)**0.6 + (80*(1-x)+20)**0.6),
[0 <= x, x <= 1])
prob.solve() # Returns the optimal value.
print("status:", prob.status)
print("optimal value", prob.value)
print("optimal var", x.value)
Prints:
status: optimal
optimal value 23.27298502822502
optimal var 0.5145387371825181

solve for elimination constant K

Can someone show me how to isolate K in the following equation? (I want to use excel to find K, and I will know a,b,c,d,and f, so I need K isolated):
a = (b/c * exp(-K*d)+a)*exp(-Kf)

Sympy Solveset Multi Variable Non-Linear Solutions

I am having a bit of trouble with Sympy's solveset. I am trying to use Sympy to find a solution to an basic circuit analysis question involving three unknown resistors and two equations. I realize that I will have to guess at the value of one of the resistors and then calculate the value of the other two resistors. I am using superposition to solve the circuit.
This is the circuit and superposition I am trying to solve
import sympy
V_outa, V_outb, R_1, R_2, R_3, V_1, V_2 = symbols('V_outa V_outb R_1 R_2 R_3 V_1 V_2')
### Here are a bunch of variable definitions.
R_eq12 = R_2*R_3/(R_2+R_3)
R_eq123 = R_eq12 + R_1
V_outa = V_1 *R_eq12/R_eq123
R_eq13 = R_1*R_3/(R_1+R_3)
R_eq123b = R_2 + R_eq13
V_outb = V_2 * R_eq13/R_eq123b
### Here is my governing equation.
V_out = 0.5* V_outb + (1/6) *V_outa
### Now I can start setting up the equations to solve. This sets the
### coefficient of the V1 term equal to the 1/2 in the governing equation
### and the coefficient of the V2 term equal to 1/6. I have also guessed
### that R_3 is equal to 10 ohms.
eq1 = Eq(1.0/2.0, V_out.coeff(V_2).subs(R_3, 10))
eq2 = Eq(1.0/6.0, V_out.coeff(V_2).subs(R_3, 10))
#### Now when I try to solve eq1 and eq2 with solveset, I get an error.
solveset([eq1, eq2], (R_2, R_3))
And here is the error I get:
ValueError: [Eq(0.500000000000000, 10*R_1/((R_1 + 10)*(10*R_1/(R_1 +
10) + R_2))), Eq(0.166666666666667, 10*R_1/((R_1 + 10)*(10*R_1/(R_1 +
10) + R_2)))] is not a valid SymPy expression
The other thing I don't understand is the set type I get when I try to solve it this way. Could someone also explain what set type this is, and how to make use of it?
expr3 = -1.0/2.0 + V_out.coeff(V_2).subs(R_3, 10)
solveset(expr3, R_2)
A screen shot of the weird set type
Any help would be much appreciated. I know it is a solvable set because Wolfram Alpha had no problems with it.
Thanks!
David

What is the approach to solve spoj KPMATRIX?

The problem link is here. The problem is basically to count all such sub matrices of a given matrix of size N by M, whose sum of elements is between A and B inclusive. N,M<=250. 10^-9<=A<=B<=10^9.
People have solved it using DP and BIT. I am not clear how.
First, i tried to solve a simpler version, 1-D case of the above problem: Given an array A, of length N, count all subarrays, where sum of elements in the subarray lies between A and B, but still couldn't think of better than O(n^2). Here is what i did :
I thought of making another array for keeping prefix sum of the original array, say prefix[N]. prefix[i] = A1 + A[2] + A[3] + ...A[i]. set prefix[ 1] = A [ 1]. Then for each i from 2 to N, problem is to count all j <= i such that sum Z = A[j] + A[j+1] + ..A[i] lies between A and B. This is equivalent to prefix[i] - prefix[j-1]. But it's still O(n^2), as for each i, j is hitting i places.
can anybody help me step by step to advance me in the given approach to solve the main problem ?.

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