Triangular Systems for the HP-41

Overview
 

-This program solves one - or several - linear system(s) whose principal matrix has only zeros above its diagonal.
-It may be used after "triangularizing" a linear system ( Gaussian method, Householder method ... )

     B = A.X    where A is a lower triangular matrix.
 

Program Listing
 
 

Data Registers:               R00 is unused                              ( Registers R01 thru Rn.m are to be initialized before executing "TRS" )

                                      •  R01 = b₁           •  Rn+1 = a_1,1      .....................      •  Rnm-n+1 = a_1,n      with  a_i,j = 0  for j > i
                                      •  R02 = b₂           •  Rn+2 = a_2,1      .....................      •  Rnm-n = a_2,n

                                       ................            ...............................................................................

                                      •  Rnn = b_n           •  R2n = a_n,1         .....................      •  Rnm = a_n,n

    >>>  At the end  x₁ , .......... , x_n   are in registers R01 to Rnn.

 •  This program may be used to solve several linear systems with the same "principal" matrix,
     especially if you want to invert the lower triangular matrix itself.

Flags: /
Subroutines: /
 
 

 
    ( 96 bytes / SIZE 1+n(c+n) )
 
 

  where  n = number of raws
    and   c = number of systems ( c + n = number of columns )

Example1:     Solve the triangular system:

   4 = 2 x
   5 = 3 x + 4 y
   6 = 5 x + 7 y + 3 z

-Store

   4    2    0    0                     R01   R04   R07   R10
   5    3    4    0        into       R02   R05   R08   R11
   6    5    7    3                     R03   R06   R09   R12

   3   ENTER^
   1   XEQ "TRS"   >>>>    ( 8 )                                    ---Execution time = 3s---

-And we have

   R01 = x =    2
   R02 = y = -1/4
   R03 = z = -3/4

Example2:    If you want to invert the above square matrix A, store the identity matrix in R01 thru R09 and A in R10 thru R18

  R01  R04  R07  R10  R13  R16      1   0   0   2   0   0
  R02  R05  R08  R11  R14  R17  =  0   1   0   3   4   0
  R03  R06  R09  R12  R15  R18      0   0   1   5   7   3

  3   ENTER^
  3   XEQ "TRS"   >>>>     ( 8 )                                    ---Execution time = 9s---

-And the inverse is in registers R01 thru R09

  R01  R04  R07       1/2       0        0
  R02  R05  R08  = -3/8      1/4      0
  R03  R06  R09      1/24  -7/12   1/3

Notes:

-The square matrix itself is unchanged
-Since synthetic registers P & Q are used, don't interrupt "TRS",
  this could change the contents of these registers.
-The alpha register is cleared at the end.

Variant:

-You can also avoid to store the zero-elements of the matrix ( above the diagonal ).
-In this case,

   add   LASTX   +   ENTER^   SIGN   ST+ X   /     after line 49
   replace line 42 by  GTO 01
   delete lines 37 to 41 and lines 14 to 16                          ( 53 lines / 90 bytes )

-In example 2, store

  R01  R04  R07  R10  R13  R15      1   0   0   2   4   3
  R02  R05  R08  R11  R14          =  0   1   0   3   7
  R03  R06  R09  R12                       0   0   1   5

  3   ENTER^
  3   XEQ "TRS"   >>>>     ( 5 )                                    ---Execution time = 8s---

-And the inverse is still in registers R01 thru R09

  R01  R04  R07       1/2       0        0
  R02  R05  R08  = -3/8      1/4      0
  R03  R06  R09      1/24  -7/12   1/3

-Since the SIZE becomes  1 + c n + n(n+1)/2  so you can solve a triangular system of 23 equations in 23 unknowns.