Question

Suppose we have a matrix A Rmxn. Recall the Golub-Kahan bidiagonalisation pro- cedure and the Lawson-Hanson-Chan (LHC) bidiagonalisation procedure (Section 8.2). Answer the following questions: (i) Workout the operation counts required by the Golub-Kahan bidiagonalisation (ii) Workout the operation counts required by the LHC bidiagonalisation. (iii) Using the ratio m, derive and explain under what circumstances the LHC is com- putationally more advantageous than the Golub-Kahan. we have a bidiagonal matrix B Rnxn, show that both B B and BB are tridiagonal matrices. Hint: recall that the operation counts of the QR factorisation (using Householder reflec- tion) is about 2mn2-In. You can relate those two bidiagonalisation procedures to the QR factorisation to work out their operation counts.

Answer 1

Let as take the matrix A = [x x x x x x x x x x x x x x x x x x x x] where matrix A elementof R^m times n House hold reflectors connected then again from the left & the correct will be utilized to zero pieces of the lattice features (i) the operation counts required by the golub - kahan bidiagonalization u_1 = [1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] A = [x x x x x x x x x x x x x x x x x x x x] U_1^* A = [x 0 0 0 0 x x x x x x x x x x x x x x x] Av, hence the right side of the top last two positions will be zero