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
 
 

01  LBL "QUAD"   02  LBL 01   03  RCL 00   04  ENTER^   05  X^2   06  +   07  2   08  /   09  RCL 00   10   E3   11  /   12  +   13  ENTER^   14  ENTER^   15  CLX   16  LBL 02   17  RCL IND Z   18  ABS   19  X>Y?   20  STO Z   21  X>Y?   22  +   23  RDN   24  DSE Z   25  GTO 02   26  VIEW X   27   E-8       28  X>Y?   29  GTO 11     30  CLX   31  ENTER^   32  R^   33  INT   34  STO M   35  RCL 00   36  -   37  X<>Y   38  LBL 03   39  ISG Z   40  CLX   41  RCL 00   42  +   43  R^   44  -   45  X<Y?

46  GTO 03   47  -   48  RCL 00   49  +   50  RCL IND X   51  RCL IND Z   52  -   53  RCL IND M   54  X<>Y   55  R-P   56  CLX   57  2   58  /   59  1   60  P-R   61  STO N   62  RDN   63  STO O   64  X^2   65  RCL IND Z   66  *   67  STO P      68  CLX   69  RCL IND Y   70  RCL N   71  X^2   72  *   73  ST+ P   74  CLX   75  X<> IND M   76  RCL N   77  *   78  RCL O   79  *   80  ENTER^   81  X<> P   82  +   83  X<> IND Y   84  RCL O   85  X^2   86  *   87  RCL P   88  -   89  X<> IND Z   90  RCL N

91  X^2   92  *   93  ST+ IND Z   94  CLX   95  RCL 00   96  ENTER^   97  X^2   98  +   99  2 100  / 101  STO P   102  ST+ Z 103  + 104  XEQ 07 105  RCL 00 106  DSE X 107  X<> P 108  ST- Z 109  - 110  LBL 04 111  RCL P 112  ST+ Z 113  + 114  RCL M 115  X=Y? 116  GTO 00 117  RDN 118  XEQ 07 119  DSE P 120  GTO 04 121  LBL 00 122  RDN 123  LBL 05 124  RCL P 125  ENTER^ 126  SIGN 127  ST+ T 128  - 129  + 130  X<>Y 131  RCL M 132  X=Y? 133  GTO 00 134  X<> Z 135  XEQ 07

136  DSE P 137  GTO 05 138  LBL 00 139  X<> Z 140  LBL 06 141  DSE P 142  X=0? 143  GTO 01  144  ISG X 145  CLX 146  ISG Y 147  CLX 148  XEQ 07 149  GTO 06 150  LBL 07 151  RCL IND X 152  RCL O 153  * 154  SIGN 155  CLX 156  RCL IND Z 157  RCL N 158  ST* Y 159  X<> L 160  - 161  X<> IND Z 162  RCL O 163  * 164  X<> IND Y 165  RCL N 166  * 167  ST+ IND Y 168  RDN 169  RTN 170  LBL 08    171  RCL 00 172  ST+ X 173   E-7 174  ISG Y 175  LBL 09 176  RCL IND Y 177  ABS 178  X<Y? 179  CLX 180  X=0?

181  STO IND Z 182  RDN 183  DSE Y 184  GTO 09 185  RTN 186  LBL 10 187  ST/ IND M 188  DSE M 189  GTO 10 190  RTN 191  LBL 11  192  RCL 00  193  ENTER^ 194  X^2 195  + 196  2 197  / 198  RCL 00 199  1 200  ST+ Z 201  + 202   E3 203  / 204  ST+ Y 205  LASTX 206  / 207  + 208  REGMOVE 209  CLD 210  XEQ 08 211  RCL 00 212  ENTER^ 213  ST+ X 214  STO M 215  ISG M 216  LBL 12 217  RCL IND Y 218  X=0? 219  GTO 12 220  4 221  * 222  RCL IND Y 223  X^2 224  X<>Y 225  /

226  ST+ IND M 227  CLX 228  STO IND Y 229  LBL 12 230  RDN 231  DSE X 232  DSE Y 233  GTO 12 234  XEQ 08 235  RCL IND M 236  X#0? 237  XEQ 10 238  RCL 00 239  STO N 240  ST+ X 241  STO M 242  ENTER^ 243  ENTER^ 244  CLX 245  ISG Z 246  LBL 13 247  RCL IND Y 248  X=0? 249  GTO 13 250  R-P 251  STO IND Z 252  X<>Y 253  CLX 254  STO IND T 255  ENTER^ 256  + 257  LBL 13 258  RDN 259  DSE Y 260  DSE N 261  GTO 13 262  X#0? 263  XEQ 10 264  CLA 265  END

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

STACK	INPUTS	OUTPUTS
X	/	/

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