Quadratic Hypersurfaces for the HP-41

Overview
 

-The following program reduces the cartesian equation of a quadratic hypersurface (QHS) in a n-dimensional space  ( n > 1 )

-Given    (QHS):  a₁₁ x₁² + a₂₂ x₂² + .............. + a_nn x_n² + Sum _{i < j}  a_{i j} x_i x_j  +  b₁ x₁ + b₂ x₂ + ............. + b_n x_n =  c

    the elements  a_{i j} ( i < j ) are gradually zeroed by the Jacobi's iterative method.

-When all these elements are smaller than 10 ^-8 ( line 27 ) , the quadratic form is diagonalized.
 ( the n eigenvalues are in registers R01 thru Rnn at this step )
-Then ( lines 211 to 265 ) several translations and/or rotations are performed to reduce the equation further.
 

Data Registers:           •  R00 = n                        ( All these registers are to be initialized before executing "QUAD"  )

           I              •  R01 = a₁₁    • Rn+1 = a_1,2   • R2n    = a_2,3   ..................................  • R(n²+n)/2 = a_n-1,n       • R(n²+n)/2+1 = b₁        • R(n²+3n)/2+1 = c
          N             •  R02 = a₂₂    • Rn+2 = a_1,3   • R2n+1 = a_2,4     ...................                                                    • R(n²+n)/2+2 = b₂
          P                ...............         .................      ...................                                                                               ........................
          U               ...............         .................    • R3n-2 = a_2,n
          T               ................      • R2n-1 = a_1,n
          S              •  Rnn = a_nn                                                                                                                              • R(n²+3n)/2 = b_n

 -----------------------------------------------------------------------------------------------------------------------------------------------------------------

                  When the program stops:

                                       R00 = n
          O
          U              R01 = a'₁₁      Rn+1 =  b'₁       R2n+1 = c'
          T               R02 = a'₂₂      Rn+2 =  b'₂
          P              ...............         .................
          U
          T
          S               Rnn = a'_nn      R2n =   b'_n

The reduced equation is:    (QHS):  a'₁₁ X₁² + a'₂₂ X₂² + .............. + a'_nn X_n²  + b'₁ X₁ + b'₂ X₂ + ............. + b'_n X_n =  c'

 where   b'_i  = 0  if  a'_ii # 0  and  c' = 0 or 1

Flags: /
Subroutines: /
 

Program Listing
 

-The synthetic registers  M , N , O , P  may be replaced by  any unused standard registers,
  for instance,  R92  R93  R94  R95  , provided  n < 13

-Lines 29-143 are three-byte GTOs
-Lines 170 to 185 replace by zero all the coefficients smaller than  10 ^-7 in magnitude ( line 173 )
-Lines 192 to 208 move the coefficients b_i & c  from registers   R(n²+n)/2+1  thru  R(n²+3n)/2+1    to registers   Rn+1  thru  R2n+1
 
 

 
   ( 397 bytes / SIZE ( n²+3n+4 )/2 )
 
 

Example:    (QHS):  3 x² + 4 y² + 7 z² + 6 t² + 9 x.y + 3 x.z + 7 x.t + 4 y.z + 8 y.t + 2 z.t + 9 x + 2 y + 5 z + 4 t = 10  in a 4-dimensional space.

       SIZE 016  or greater

   4  STO 00  and we store these ( n²+3n+2 )/2 = 15 coefficients as shown below:

           3     9     4      2       9       10                      R01     R05     R08     R10      R11       R15
           4     3     8               2                     in          R02     R06     R09                  R12                     respectively
           7     7                      5                                  R03     R07                             R13
           6                             4                                  R04                                         R14
 

 XEQ "QUAD"  the HP-41 displays the successive greatest  | a_{i j} |  ( which tend to zero )  and when the program stops,
  the results are in registers R01 thru R2n+1

         R01 =  -0.209131158     R05 = 0      R09 = 1                       Execution time = 2mn28s
         R02 =   0.297006169     R06 = 0
         R03 =   1.235467092     R07 = 0
         R04 =   2.716166061     R08 = 0

  So, the reduced equation is   -0.209131158 X² +  0.297006169 Y² + 1.235467092 Z² + 2.716166061 T² = 1    a "hyper-hyperboloid"?
 

Notes:

-The 4 eigenvalues of the initial quadratic form are     -1.035428817  ,   1.470506591  ,  6.116918408  ,  13.44800383
-These 4 numbers are in registers R01 thru R04  when the program reaches line 191 or line 209

-If you want to save these values, for instance in registers R2n+2 to R3n+1,

      add  RCL 00   RCL 00  .1   %   +   +   2   +   E3   /    1   +   REGMOVE   after line 209 or 210

-In the above example, n = 4 and the eigenvalues will be eventually in registers  R10 R11 R12 R13

-If you want to compute the n eigenvalues only,

   store  n into R00 , the coefficients a_{i i} & a_{i j}  into R01 thru  R(n²+n)/2  and
   replace lines 170 to 265  by  LBL 11   CLA  CLD  END
   replace lines 95 to 109 by  RCL 00   DSE X   STO P  ( synthetic )  RDN           >>  They will be in R01 thru Rnn at the end.

-Actually, these eigenvalues are the n eigenvalues of the symmetric matrix  M = [ m_{i j} ]  defined by   m_{i i} = a_{i i}  and  m_{i j} = m_{j i} = (1/2) a_{i j}  for i # j

-The results are quite similar to those given by "QUADRIC"  in a 3-dimensional space ( cf "Quadrics for the HP-41" )
  but here, the names of the hypersurfaces have been omitted for 2 reasons:

    1°) This would require many extra bytes.
    2°) I don't know the exact terminology !

-For n = 7 , the execution time is of the order of  18 minutes.
-If this program is executed from extended memory, n could reach 23.
 

Exercise:    (QHS):   2 x² + 3 y² + 4 z² + 2 t² + 2 x.y + 6 x.z + 4 x.t + 12 y.z + 2 y.t + 6 z.t + 2 x + 3 y + 4 z + 5 t = 10  in a 4-dimensional space.

Answer:    The reduced equation is:     Y = 1.461929785 X² -  1.113606716 Z² -  5.533772793 T²    a kind of hyperbolic hyperparaboloid ?

  and the 4 eigenvalues of the original quadratic form are:    -3.101221397 ,  0  ,  2.362316584  ,  11.73890481