Pentadiagonal solver Write a MATLAB function penta below that solves pentadiagonal systems of equations of size n. A pentadiagonal matrix is a banded one with bandwidth = 5 and p = q = 3. Assume that no pivoting is needed, but do not assume that the matrix A is symmetric. Your program should expect as input six vectors: md = main diagonal vector of A (length n) ld = lower diagonal of A (length n - 1) . lld = next lower diagonal of A (length n - 2) ud = upper diagonal of A (length n - - 1) uud = next upper diagonal of A (length n 2) b = right hand side vector (length n) It should calculate and return x = A-'b using a Gaussian elimination variant that required O(n) flops and consumes no additional space as a function of n (i.e., in total 7n storage locations are required). Try your program on the matrix defined by n = 100,000, dj,i+2 = di+2,i = i, di,i+1 = dj+1,i = 2i, and aj,i = 8i, for all i such that the relevant subscripts fall in the range 1 to n. Derive a right hand side vector b = Axexact using Xexact = = (1, 1, ..., 1)? Then solve for x given this b and record || Xexact -x||2 to verify your solver. function [x, md, ud, uud] = penta (md, ld, ud, lld, uud, b) % Solve Ax = b for a pentadiagonal A md = main diagonal vector of A (length n) ld lower diagonal of A (length (n-1)) ild = next lower diagonal of A (length (n-2)) ud upper diagonal of A (length (n-1)) uud = next upper diagonal of A (length (n-2)) b right hand side vector (length n) = end % penta Pentadiagonal solver Write a MATLAB function penta below that solves pentadiagonal systems of equations of size n. A pentadiagonal matrix is a banded one with bandwidth = 5 and p = q = 3. Assume that no pivoting is needed, but do not assume that the matrix A is symmetric. Your program should expect as input six vectors: md = main diagonal vector of A (length n) ld = lower diagonal of A (length n - 1) . lld = next lower diagonal of A (length n - 2) ud = upper diagonal of A (length n - - 1) uud = next upper diagonal of A (length n 2) b = right hand side vector (length n) It should calculate and return x = A-'b using a Gaussian elimination variant that required O(n) flops and consumes no additional space as a function of n (i.e., in total 7n storage locations are required). Try your program on the matrix defined by n = 100,000, dj,i+2 = di+2,i = i, di,i+1 = dj+1,i = 2i, and aj,i = 8i, for all i such that the relevant subscripts fall in the range 1 to n. Derive a right hand side vector b = Axexact using Xexact = = (1, 1, ..., 1)? Then solve for x given this b and record || Xexact -x||2 to verify your solver. function [x, md, ud, uud] = penta (md, ld, ud, lld, uud, b) % Solve Ax = b for a pentadiagonal A md = main diagonal vector of A (length n) ld lower diagonal of A (length (n-1)) ild = next lower diagonal of A (length (n-2)) ud upper diagonal of A (length (n-1)) uud = next upper diagonal of A (length (n-2)) b right hand side vector (length n) = end % penta
