the alp chapter three widget problem is explicitly solved in tableau form with four iterations. every operation’s easily checked against code. the target is 7094 f66 using btran and ftran from tape.

at first, the incoming column and pivot row numbers for each iteration are simply taken ‘as is’ from alp chapter three for each iteration. this makes the skeletal mechanism of the ‘revised’ simplex more apparent. for example, it’s possible to say that the first column of the tableau is untouched. it will become apparent later that the ‘pricing’ mechanism for choosing incoming columns does indirectly touch column one, but at first this can be swept under the carpet.

smpl1 - alp chapter three example, worked exactly as it appears in the book. pre ‘revised’ simplex. the entire tableau is updated in each iteration. column one of the tableau is a unit vector that never directly plays a role in the four iterations needed, but it’s touched in each iteration, with potential numerical issues.

smpl2 - ‘revised’ simplex, chapter four of alp. the full tableau is not updated, the original serves as a kind of constant. instead, the beta column ‘right hand side’ is updated using the basis inverse. new eta vectors are generated and ready for ‘storage on tape’. each eta is a column of the basis inverse. the basis inverse is also used to transform incoming columns, which has the same end effect as the ‘tape based’ ftran routine. the first column is never touched.

smpl3 - tape based ftran for transforming incoming columns. tape is emulated by storing pivot value and eta column pairs left to right in a matrix. this revised simplex mechanism is approaching what was used in the late fifties and early sixties. only the ‘right hand side’ beta vector is maintained across iterations. the basis inverse is stored as eta vectors ‘on tape’.

a notable curiosity here is the relationship between tape storage of eta columns and descendant ‘list data structures’. ftran is an ancestor of the ‘python list’. there’s a sequential data container, filled ‘first in first out’ with vectors. ftran iterates over the tape ‘forwards’, and btran iterates over the tape ‘backwards’.