Let f(x) = x¹ Hx-x¹b, where H and b are constant, independent of x, and H is symmetric positive definite. Given vectors x0) and p0), find the value of the scalar a that minimizes f(x0) + ap0)). This is the formula for the stepsize ak in the linear conjugate gradient algorithm.
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- USING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.Determine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)Explain the Wronskian determinant test. Using the Wronskian determinant test, write the program using NumPy to determine whether the functions f(x)=e^(- 3x), g(x)=cos2x and h(x)=sin2x are linearly independent in the range (-∞, + ∞). #UsePython
- for linear algebra in solving a sytem of linear equations in form AX = B. True or false. 1.the array of unknown x be the same size as array of constants in B 2. the number of rows of A must be the same as rows of B 3. A must have the same number of columns as rows 4. Number of columns in A must be the same as number of rows in BLet G: (0,1}" {0,1)"+l be defined as follows: G(x....x) = (X1, X Ox2, X O x2 O X3, ..., X1 O X2 O ... O x X). Prove or G is a PRG. disproveWrite pseudocode for an algorithm for finding the real roots of equation ax2 + bx + c for arbitrary real coefficients a, b, and c. (You may assume the availability of the square root function sqrt(x).)
- suppose a computer solves a 100x100 matrix using Gauss elimination with partial pivoting in 1 second, how long will it take to solve a 300x300 matrix using Gauss elimination with partial pivoting on the same computer? and if you have a limit of 100 seconds to solve a matrix of size (N x N) using Gauss elimination with partial pivoting, what is the largest N can you do? show all the steps of the solutionAnalyze the running time (i.e. T(n)) of these functions. You should be able to find some simple function f(n) such that T(n) O(f(n)). You should show your work and rigorously justify your an- 1. swer.Prove that f(x) = x is O(x3).
- Let f(n) and g(n) be asymptotically positive functions. Prove or disprove following. f(n) + g(n) = q(min(f(n), g(n))).Pls Use Python Explain the Wronskian determinant test. Using the Wronskian determinant test, write the program using NumPy to determine whether the functions f(x)=e^(- 3x), g(x)=cos2x and h(x)=sin2x are linearly independent in the range (-∞, + ∞). Ps: Please also explain step by step with " # "Program the Gaussian elimination method with no partial pivoting for solving a linear system of the form Ax=b, where b is a single column vector. Your function should take in a coefficient matrix A, and a single input vector b. Your function should return the solution vector x. Your code should also return the appropriate error message. The first line of your function should look like: function x = gaussElimination (A,b)