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HP Forum Archive 21

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Matrix Characteristic Polynomial - Reloaded.
Message #1 Posted by Ángel Martin on 19 Aug 2013, 9:02 a.m.

Just to keep you all updated, the SandMatrix module is close to release. While preparing the manual I’ve been tweaking and improving some of the routines, taking advantage of its extended function set. One of these has been CHRPOL, used to obtain the characteristic polynomial of a matrix.

The new version code is about 1/3rd. of the original one, and does the job in about half the time – if not faster. Such is the improvement derived from the “direct approach”. An image is worth 1,000 words, so here’s the program listing for you.- Stay tuned, the real think will be packed…

Magic in 60 program steps.

Uses the Faddevv-Leverrier method - you can almost read the algorithm from the ALPHA register, directly using the abstraction layer of the Matrix functions. 

1	LBL "CHRPOL"	MNAME in Alpha
2	DIM?    	n,00n
3	I#J?	        not square?
4	-ADV MATRX	show error
5	ASTO 01	        MNAME
6	-CCD MATRIX	shows 'RUNNING…" 
7	"|-,P"	        requires auxiliary array
8	MAT=    	P = A
9	ASWAP	        swap alpha around comma
10	DIM?    	n,00n
11	INT     	n 
12	E	
13	+       	n+1
14	MDET    	independent term
15	STO IND Y	stored in Rn+1
16	ASWAP	        swap again
17	MAT=    	avoids LU issues
18	DIM?	
19	"#"     	another auxiliary array
20	MATDIM	
21	FRC     	0,00n
22	2	
23	+       	2,00n
24	STO 00	
25	CF 21   	not halting VIEW
26	LBL 00	
27	VIEW 00  	shows current index
28	"#"	
29	MIDN    	[#] = [I]
30	"P"	
31	MTRACE   	tr(P)
32	RCL 00	
33	INT     	k+1
34	E	
35	-       	k 
36	/	
37	CHS	
38	STO IND 00	pk = -tr(P) / k
39	"X,#,#" 	
40	MAT*    	[#] = pk [I]
41	"P,#,#" 	
42	MAT+    	[#] = [P] + p[ I]
43	CLA	
44	ARCL 01	
45	"|-,#,P"	
46	M*M     	B= A (B - p I)
47	ISG 00	        increment counter
48	GTO 00	        next coefficient
49	DIM?    	n,00n
50	FRC     	0,00n
51	E	
52	STO 01   	it's monic (!)
53	E3/E+   	1.001
54	+       	1.00(n+1) - cnt'l word
55	"#"	
56	PURFL	        remove auxiliary arrays
57	"P"	
58	PURFL	
59	MNAME?  	bring MNAME back
60	END     	done

The polynomial control word is left in X upon termination - that is, registers containing the n+1 coefficients in the form bbb.eee -from Rbbb to Reee.

Cheers, ÁM

PS. Here’s an article that may be useful to jog your memory:
http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/articles.cgi?read=944

Edited: 19 Aug 2013, 9:10 a.m.

      
Re: Matrix Characteristic Polynomial - Reloaded.
Message #2 Posted by Massimo Gnerucci (Italy) on 19 Aug 2013, 10:01 a.m.,
in response to message #1 by Ángel Martin

            
Re: Matrix Characteristic Polynomial - Reloaded.
Message #3 Posted by Ángel Martin on 19 Aug 2013, 1:23 p.m.,
in response to message #2 by Massimo Gnerucci (Italy)

LOL !!! You gotta love this forum :)

A suitable ovation for what may well be "the last program ever to be written using the Advantage-based tool-set".

Cheers, 'AM

                  
Re: Matrix Characteristic Polynomial - Reloaded.
Message #4 Posted by Massimo Gnerucci (Italy) on 19 Aug 2013, 1:55 p.m.,
in response to message #3 by Ángel Martin

Quote:
LOL !!! You gotta love this forum :)

I really do!

Besides you deserve our applause for your continued work of love on Mcode routines.

Thank you Ángel.

Massimo

            
Re: Matrix Characteristic Polynomial - Reloaded.
Message #5 Posted by Namir on 19 Aug 2013, 2:24 p.m.,
in response to message #2 by Massimo Gnerucci (Italy)

Hey!! What are MY PARENTS doing in the first row of that picture!!!??????????!!!!!???????????!!!!???

:-)

Namir

                  
Re: Matrix Characteristic Polynomial - Reloaded.
Message #6 Posted by Jim Horn on 19 Aug 2013, 3:32 p.m.,
in response to message #5 by Namir

And why are they sitting with my kids?

      
Re: Matrix Characteristic Polynomial - Reloaded.
Message #7 Posted by Monte Dalrymple on 19 Aug 2013, 4:54 p.m.,
in response to message #1 by Ángel Martin

'Angel, do you have a rough ETA? I'm putting together a new Flash image for the 41CL.

Thanks, Monte

            
Re: Matrix Characteristic Polynomial - Reloaded.
Message #8 Posted by Ángel Martin on 20 Aug 2013, 9:29 a.m.,
in response to message #7 by Monte Dalrymple

I'm hoping for end of this week, just a couple of wrinkles left. I'll send you the ROM images as soon as they're ready, won't wait for the manual (which is another thick one).

      
Re: Matrix Characteristic Polynomial - Reloaded.
Message #9 Posted by Robert Prosperi on 19 Aug 2013, 8:18 p.m.,
in response to message #1 by Ángel Martin

Angel:

Quote:
Uses the Faddevv-Leverrier method

Wow, a blast from my very distant past! In college, I had a Mech Eng & Numerical Analysis Professor that seemed to be able to prove virtually anything he chose, using Feddevv-Leverrier. Always mentioned that Faddevv was married to Faddevva (sp?) and that they worked on these matrix equations together. Such Marital bliss...

Were it not for your post I may have believed I imagined it all. Thanks for the mention. It's sometimes odd how these posts will hit you.

--bob prosperi

      
Re: Matrix Characteristic Polynomial - Reloaded.
Message #10 Posted by Thomas Klemm on 21 Aug 2013, 8:24 a.m.,
in response to message #1 by Ángel Martin

For those interested here's a HP-15C version of the algorithm:

Listing

001 - 42,21,11  LBL A             
002 - 45,23,11  RCL DIM A         ; -- n  n
003 -        1  1                 ; -- n  n  1
004 - 42,23,14  DIM D             ; D(n, 1)
005 - 44,16,14  STO MATRIX D      ; D = [1, ..., 1]
006 -    44  0  STO 0             ; a0 = 1
007 -       34  x<>y              ; -- n  1  n
008 -       26  EEX               
009 -        3  3                 ; -- n  1  n  1000
010 -       10  /                 ; -- n  1  0.00n
011 -       40  +                 ; -- n  1.00n
012 -    44 25  STO I             ; I = 1.00n
013 - 45,16,11  RCL MATRIX A      
014 - 44,16,12  STO MATRIX B      ; B = A
015 - 42,21, 0  LBL 0             
016 - 42,26,13  RESULT C          
017 - 45,16,11  RCL MATRIX A      
018 - 45,16,12  RCL MATRIX B      
019 -       20  x                 ; C = A * B
020 - 45,23,12  RCL DIM B         ; -- n  n
021 -        1  1                 ; -- n  n  1
022 -       40  +                 ; -- n  n+1
023 - 42,23,12  DIM B             ; B(n, n+1)
024 - 45,16,12  RCL MATRIX B      ; -- n  n+1  B(n, n+1)
025 - 42,16, 4  MATRIX 4          ; -- n  n+1  B(n+1, n)
026 -        1  1                 ; -- n  n+1  B(n+1, n)  1
027 -    43 33  R^                ; -- n+1  B(n+1, n)  1  n
028 - 42,23,12  DIM B             ; B(1, n)
029 - 42,26,15  RESULT E          
030 - 45,16,12  RCL MATRIX B      ; -- B(1, n)
031 - 45,16,14  RCL MATRIX D      ; -- B(1, n)  D(n, 1)
032 -       20  x                 ; E = B x D
033 - 42,16, 9  MATRIX 9          ; p = |E| = Tr(B)
034 -       16  CHS               ; -- -p
035 -    45 25  RCL I             ; -- -p  k.00n
036 -    43 44  INT               ; -- -p  k
037 -       10  /                 ; -- -p/k
038 -    44 24  STO (i)           ; -- ak = -p/k
039 - 42,26,12  RESULT B          
040 - 45,16,11  RCL MATRIX A      
041 -       20  x                 ; -p/k * A
042 - 45,16,13  RCL MATRIX C      
043 -       40  +                 ; B = C -p/k * A
044 - 42, 6,25  ISG I             
045 -    22  0  GTO 0

Calculating the Trace of a Matrix

       DIM(n, n+1)      AT         DIM(1, n)                     DET

|* . .|      |* . . .|      |* * *|                |1|
|. * .|  ->  |* . . .|  ->  |. . 0|  ->  |* * *| x |1|  ->  |*|  ->  *
|. . *|      |* 0 0 0|      |. . 0|                |1|
                            |. . 0|

Example:

3
ENTER
DIM A
MATRIX 1
USER
3 STO A
1 STO A
5 STO A
3 STO A
3 STO A
1 STO A
4 STO A
6 STO A
4 STO A
GSB A
RCL 0 -> 1
RCL 1 -> -10
RCL 2 -> 4
RCL 3 -> -40

Thus the characteristic polynomial is: p(x) = x3 - 10x2 + 4x - 40.

Eigenvalues

We can use this short program to solve p(x) = 0. 
LBL B
RCL+ 1
*
RCL+ 2
*
RCL+ 3
RTN

CLx
SOLVE B
-> 10.0000
The other solutions are 2i and -2i which can't be found with the built-in solver.

Cheers
Thomas

Edited: 21 Aug 2013, 4:39 p.m.

            
Re: Matrix Characteristic Polynomial - Reloaded.
Message #11 Posted by Ángel Martin on 22 Aug 2013, 2:14 p.m.,
in response to message #10 by Thomas Klemm

Nice tricks Thomas! Would you also have one to create an identity matrix - the quickest possible way?

Also if you're going to end up using the SOLVEr, might as well have started there, solving for x in the equation:

Det( A - x I) = 0

Cheers, 'AM

                  
Re: Matrix Characteristic Polynomial - Reloaded.
Message #12 Posted by Thomas Klemm on 22 Aug 2013, 4:48 p.m.,
in response to message #11 by Ángel Martin

If we know that A is invertible: 

RESULT E
RCL MATRIX A
ENTER
/

Otherwise something along this: 

       DIM(n+1, n)    AT           DIM(n, n)

             |1 1 1|      |1 0 0 0|      |1 0 0|
|1 1 1|  ->  |0 0 0|  ->  |1 0 0 0|  ->  |0 1 0|
             |0 0 0|      |1 0 0 0|      |0 0 1|
             |0 0 0|

Kind regards
Thomas

                  
Re: Matrix Characteristic Polynomial - Reloaded.
Message #13 Posted by Thomas Klemm on 22 Aug 2013, 5:33 p.m.,
in response to message #11 by Ángel Martin

Quote:
Det( A - x I) = 0

That's actually a nice idea: 

LBL E
RESULT B
RCL MATRIX E
*
RCL MATRIX A
-
MATRIX 9
RTN

0
SOLVE E
-> 10

Cheers
Thomas

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