Faster factor finder method for 42S
11-26-2019, 04:24 PM
Post: #1
 Dave Britten Senior Member Posts: 2,074 Joined: Dec 2013
Faster factor finder method for 42S
There's nothing novel here from a mathematics standpoint - it's just the same old mod-30 sieve we've been doing for decades. Rather, I found a way to use matrices on the 42S to do the trial divisions much faster.

Here's the code with some C-like comments:

Code:
LBL "FACTOR" XEQ "SS"    //Save stack //Store number to factor, and its square root STO 01 SQRT STO 04 //Initialization CLA SF 01 CF 09 //Create the matrix for the first 4 divisors 4 1 DIM "D" INDEX "D" J- 0 2 XEQ 00 1 XEQ 00 2 XEQ 00 2 XEQ 00 STO 00 //Main loop LBL 01 FS? 09        //Completely factored? GTO 09        //Go to exit routine. //Divide the remaining product by all divisors and store fractional parts in "R" RCL 01 RCL "D" ÷ FP STO "R" //See if any remainder is 0 INDEX "R" [MIN] X=0? GTO 02 //See if the current max divisor is greater than the square root RCL 04 RCL 00 X>Y? GTO 06 //First time through the loop, go set up the matrix for the next 8 divisors FS?C 01 GTO 10 //Otherwise add 30 to all divisors for the next loop 30 STO+ "D" STO+ 00 GTO 01        //Back to the start of the loop. //Main loop ends here. //Last factor is the remaining product LBL 06 RCL 01 AIP AVIEW GTO 09 //Factor out the current divisor LBL 02 RDown        //Get the divisor that corresponds to the zero remainder from trial division 1 INDEX "D" STOIJ RCLEL STO 02 CLX        //Initialize factor's power to zero STO 03 LBL 03        //Divide out factor as many times as possible RCL 01 RCL 02 MOD X≠0? GTO 04 LASTX STO÷ 01 1 STO+ 03 GTO 03 LBL 04        //Display update routine RCL 02 AIP 1 RCL 03 X>Y? ├"↑" X>Y? AIP 1        //Check if fully factored RCL 01 X=Y? SF 09 SQRT        //Update the square root of remaining product STO 04 FC? 09        //Append "×" if there are more factors ├"×" AVIEW GTO 01        //Go make another pass with the same set of divisors //Add the next divisor to the divisor matrix using increment in X LBL 00 + STOEL J- RTN //Create the 8-element divisor matrix for passes 2+ LBL 10 8 1 DIM "D" INDEX "D" J- RCL 00 4 XEQ 00 2 XEQ 00 4 XEQ 00 2 XEQ 00 4 XEQ 00 6 XEQ 00 2 XEQ 00 6 XEQ 00 STO 00 GTO 01        //Back to the start of the loop. //Exit & cleanup LBL 09 CF 01 CF 09 XEQ "RS"    //Restore stack END

"SS" and "RS" are the save-stack and restore-stack routines I have on my 42S. You can remove/substitute these calls as needed.

Variables, registers, and flags:

Code:
Regs 00 - Current max divisor 01 - Number to factor/remaining product 02 - Current divisor/factor 03 - Power of current factor 04 - Square root of number to factor/remaining product Flags 01 - First pass 09 - Fully factored Variables "D" - Divisors "R" - Remainders

The basic idea is that rather than repeatedly doing 4, XEQ 01, 2, XEQ 01, 4 XEQ 01... making 8 calls in each loop, all the trial divisors are stored in a matrix and divided all at once. Then FP takes their fractional parts, and the undocumented matrix function [MIN] identifies any of the divisions with no remainder. Found factors get fully divided out and displayed, and if none are found, 30 is added to the divisor matrix and the loop repeats. This continues until the divisors exceed the square root of the remaining product, or the input number has been fully factored down to 1.

A couple of informal timing tests of factoring 9,999,999,971:

Original mod-210 factor finder with incr. XEQ - 1m30s
New matrix-based mod-30 factor finder - 53s

I got the idea from a TI-84 factor finder that uses essentially the same technique with lists. It dawned on me that it could be done with matrices on the 42S.

The code is still kind of rough and unrefined, but I'm seeing execution times cut as much as in half for certain inputs. I haven't yet tried extending this into a full mod-210 algorithm, so there may be more speed gains to be had yet. It would require larger divisor and remainder matrices, though, so more memory would be required to run the program.
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 Messages In This Thread Faster factor finder method for 42S - Dave Britten - 11-26-2019 04:24 PM RE: Faster factor finder method for 42S - Csaba Tizedes - 11-29-2019, 09:38 AM RE: Faster factor finder method for 42S - Dave Britten - 11-29-2019, 01:26 PM RE: Faster factor finder method for 42S - Paul Dale - 11-29-2019, 10:16 AM RE: Faster factor finder method for 42S - grsbanks - 11-29-2019, 11:27 AM RE: Faster factor finder method for 42S - Joe Horn - 11-29-2019, 01:12 PM RE: Faster factor finder method for 42S - grsbanks - 11-29-2019, 01:25 PM RE: Faster factor finder method for 42S - Joe Horn - 11-29-2019, 02:04 PM RE: Faster factor finder method for 42S - Dave Britten - 11-29-2019, 04:27 PM RE: Faster factor finder method for 42S - DMaier - 11-30-2019, 09:32 PM RE: Faster factor finder method for 42S - J-F Garnier - 12-01-2019, 08:02 PM RE: Faster factor finder method for 42S - Werner - 12-02-2019, 07:17 AM RE: Faster factor finder method for 42S - Dave Britten - 12-02-2019, 12:19 PM

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