Create

$[$ S$,$ U $,$ V $]=$ SVDBiDiagImpShift$($ B $)$

$1$

to calculate the SVD of a bidiagonal matrix via the algorithm described in $11\mathrm{.}2\mathrm{.}4.$ Rather than a matrix of mostly zeros, the singular values should be returned in a single vector, S$.$

Do not perform full matrix multiplications when updating B$,$ U$,$ and V$,$ only modify the parts of those matrices which are affected by the current givens rotation you$$re applying.

Here$$s an example of matlab syntax that will be helpful for avoiding full matrix multiplications: A$(2$:$3,4$:$6)$ in Matlab refers to the submatrix with entries $(2,4),(2,5),(2,6)$ for the first row and $(3,4),(3,5),(3,6)$ for the second row $($starting indexing at $1,$ as Matlab does$).$ To address, for example, the entire second column, you use A$($:$,2).($There is no particular meaning to this example, it$$s just meant to illustrate Matlab syntax.$)$

Also, do not use the Francis step routine provided for the week $10$ home$-$ work, as this was designed to only update the lower triangular part of the matrix, and using that routine would unnecessarily complicate your code.

Don$$t forget to incorporate a shift at the beginning of each iteration. A choice of shift can be found in $10\mathrm{.}3\mathrm{.}6.$ You should also implement the stopping criteria described in section $10\mathrm{.}3\mathrm{.}6,$ and also include some large amount of maximum iterations in case that stopping criteria is not met.

Also, don$$t forget to reorder the columns and rows of S$,$ U$,$ and V at the end of your algorithm to match the ordering requirements of the SVD$.$

We recommend using matlab$$s built in SVD functions to test your code.