thereforeby evolving the recursive relationships up until pwe obtain- - | | - - - - = > - - > - - = these expressions will allow us to easily compute - - and | | provided the following quantities are availableckj : - - sinceby theorem we can write- - - ak- - - - ak- ak- ak- > - - - ak- jthe quantities {ckj can be computed from the recursionc - jj > - ck- ckj ck- ck- , - > - ck- , - pj kp ( observe that this recursive computation takes ( ntime and that once {ckj are availablewe can express - and |aas- - | | jj >jj = > jj > jj = in summarywe have proved the following theorem sherman-morrison recursion the inverse and determinant of the matrix respectively bypp = ak ak are given - - - cd det(adet( det( )where rnxp and pxp are the matrices : , , :diag > , > , and all the { , are computed from the recursion ( in ( ntime