Valentin's submission "Re: Your 15C results are *wrong*!" of 6:39 on July 1 ended with the following statement: "A final advice, if you would: as I understand you're working in some project re the friendly HP-TI competitions, it would be most proper to thoroughly test and validate each claimed program running times, results, and the like, made by both TI and HP sides, lest you'll find yourself embarrassed by non-reproducible, provably erroneous results. Unlikely perhaps, but a rigorous approach is mandatory for scholar-type projects." Shortly after that submission he realized that he had not reproduced my results because he was not working the same problem. As a result of his comment I belatedly realized that I have not addressed running times in my discussions. So, here goes, and if readers don't agree with my results or conclusions I am confident that they will let me know.

In my mind we need to focus on the total solution time including both the time to enter the test matrix, the running time, where running time includes the time to calculate the determinant if that is required as with the HP-41 Math Pac and the TI-59 Master Library, and the readout time. As a place to begin I used the 4x4 inversion from Kahan's paper "Mathematics Written in Sand". With the test matrix in place the approximate times for data entry and run times in seconds until indication that the inversion was complete are

Machine/Program Entry Run Totalwhere the long entry times for the HP-41 programs are due to the need to wait for the display to settle after one entry before making the next entry. If the user goes too fast he will get false entries. For the HP-28S and the TI-59 I simply can't enter the data so quickly that I can induce false entries. I am not sure what the HP-41 programs are doing during data entry. Years ago there were some TI-59 matrix inversion programs that did some processing in response to each input before accepting the next input. That does not appear to be the case with the HP-41 programs since I can recall the entered input data from memory after completing the input. Perhaps the delay is due to generation and display of prompting messages. If you know, let me know. My method for the HP-28S is to establish a 4x4 matrix by the 4 ENTER IDN sequence and enter the elements of the matrix with PUTI commands.HP-15C (from Kahan's paper) 40 11 51

HP-41 Math Pac MATRIX 65 40 105

HP-41 Advanced Solutions MATRX 65 7 72

HP-28S (/ or INV) 45 2 47

TI-59 Master Library (ML-02) 35 90 125

I did not include the times required to read out the solution in the table above. There is some short, fraction-of-a-second running time between display of individual elements for the HP-41 and TI-59 solutions. For the TI-59 there is an additional two second delay as the output switches from the end of one column to the next. That delay is used in decoding the pivoting index to identify which column to display next. For the HP-41 Math Pac program the delay when moving from one column to the next is over fifteen seconds. I think that time is actually used to calculate the elements in the next column. So far as I can tell the program delivers one column of solution elements at a time to R35 through R38. If you disagree, let me know.

There are also differences in the availability of the solution elements for further calculations. The HP-41 Advanced Solutions (MATRX) program leaves them in order, row by row, starting at R01. The TI-59 ML-02 leaves them in columns starting at R08, but the order of the columns may be scrambled if pivoting was required during the solution. The HP-28S leaves them in a matrix in level 1 where they can be operated on as desired. As noted above I'm not sure exactly what the HP-41 Math Pac solution leaves in memory