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Next: Operations on univariate polynomials Up: An introduction to Computer Algebra Previous: Operations on real numbers

Operations on matrices

AXIOM SESSIONscale=2.5]

N := PositiveInteger
 

   (1)  PositiveInteger
                                                                 Type: Domain
                                                                  Time: 0 sec

D := DoubleFloat
 

   (2)  DoubleFloat
                                                                 Type: Domain
                                                                  Time: 0 sec

F := Float
 

   (3)  Float
                                                                 Type: Domain
                                                                  Time: 0 sec

Q := Fraction Integer
 

   (4)  Fraction Integer
                                                                 Type: Domain
                                                                  Time: 0 sec


HilbertMatrixD(n:N): Matrix  D == _
  matrix([[1.0/(i + j) for i in 1..n] for j in 1..n])
 
                                                                   Type: Void
                                                   Time: 0.01 (IN) = 0.01 sec

HilbertMatrixF(n:N): Matrix  F == _
  matrix([[1.0/(i + j) for i in 1..n] for j in 1..n])

                                                                   Type: Void
                                                   Time: 0.01 (IN) = 0.01 sec

HilbertMatrixQ(n:N): Matrix  Q == _
  matrix([[1/(i + j) for i in 1..n] for j in 1..n])

                                                                   Type: Void
                                                                  Time: 0 sec


HilbertMatrixD(3)
 

        +       0.5         0.333333333333333        0.25       +
        |                                                       |
   (8)  |0.333333333333333        0.25                0.2       |
        |                                                       |
        +      0.25                0.2         0.166666666666667+
                                                     Type: Matrix DoubleFloat
                           Time: 0.04 (IN) + 0.03 (OT) + 0.01 (GC) = 0.08 sec

HilbertMatrixQ(3)
 

        +1  1  1+
        |-  -  -|
        |2  3  4|
        |       |
        |1  1  1|
   (9)  |-  -  -|
        |3  4  5|
        |       |
        |1  1  1|
        |-  -  -|
        +4  5  6+
                                                Type: Matrix Fraction Integer
                                                   Time: 0.02 (OT) = 0.02 sec

HilbertMatrixF(3)
 

   (10)
   +          0.5            0.3333333333 3333333333           0.25          +
   |                                                                         |
   |0.3333333333 3333333333           0.25                      0.2          |
   |                                                                         |
   +         0.25                      0.2            0.1666666666 6666666667+
                                                           Type: Matrix Float
                           Time: 0.01 (IN) + 0.01 (OT) + 0.01 (GC) = 0.03 sec


[determinant(HilbertMatrixD(5*n)) for n in 1..6]
 

   (11)
   [1.4878155287776663e-14, 1.1708847390332544e-58, - 7.329551900980472e-127,
    1.8972182407450097e-202, - 1.7767073603611213e-280, 0.0]
                                                       Type: List DoubleFloat
                           Time: 0.43 (IN) + 0.14 (EV) + 0.06 (OT) = 0.63 sec

[determinant(HilbertMatrixF(5*n)) for n in 1..6]
 

   (12)
   [0.1487815528 7758289063 E -13, 0.1171371536 8024357091 E -57,
    0.8781685580 3800165504 E -132, 0.4005398450 3281293592 E -224,
    - 0.8672044362 0589913695 E -319, 0.1929750883 9978983829 E -417]
                                                             Type: List Float
                                       Time: 0.48 (EV) + 0.10 (GC) = 0.58 sec

[determinant(HilbertMatrixQ(5*n)) for n in 1..6]
 

   (13)
           1                                     1
   [--------------, ----------------------------------------------------------,
    67212633600000  8537000898240926708833515201784986712482596782080000000000

       1
    /
      146538740183666877481333971725269067959384123502368457086504025547136338_
       7281567173259185477862791557143607640064000000000000000000000
     ,

       1
    /
      327723880133278729476665890222832025204546816520485699917379710978406689_
       52779223214483748651170263631423109857464822124939490164270836312516034_
       15239472195372333133701896827240771248538377176401848565760000000000000_
       00000000000000000000000
     ,

       1
    /
      943443458848441228956693696279760730359903469457936499731851074111615233_
       14255514417225236246776078681522617448130111877369572484796897612065933_
       52237684747328030203084672090167556606865123046915733557035025165367211_
       81161075206120441411355364487536402146184314196057700767052024312362680_
       20442693962925170374671475057413324800000000000000000000000000000000000_
       000000000000000
     ,

       1
    /
      347678097173631327706181116037484788665140455596751194236374423476934278_
       72336185915903600949727565688512425687551728790953230983654612947880686_
       64034084877513911919412368602956835751833985175217203858528427028478411_
       07420065708261775033449064671483311289619473162541454992338969245601369_
       24601181973306894033338133775362887249764402725843635016814711320344415_
       32672272163948431375908684874930724465894995883925403831087253290653960_
       68715359788388174845939833221273153240082092406289203200000000000000000_
       00000000000000000000000000000000000000
     ]
                                                  Type: List Fraction Integer
                                       Time: 0.31 (EV) + 0.01 (OT) = 0.32 sec



h6F := HilbertMatrixF(6)
 

   (14)
   [
     [0.5, 0.3333333333 3333333333, 0.25, 0.2, 0.1666666666 6666666667,
      0.1428571428 5714285714]
     ,

     [0.3333333333 3333333333, 0.25, 0.2, 0.1666666666 6666666667,
      0.1428571428 5714285714, 0.125]
     ,

     [0.25, 0.2, 0.1666666666 6666666667, 0.1428571428 5714285714, 0.125,
      0.1111111111 1111111111]
     ,

     [0.2, 0.1666666666 6666666667, 0.1428571428 5714285714, 0.125,
      0.1111111111 1111111111, 0.1]
     ,

     [0.1666666666 6666666667, 0.1428571428 5714285714, 0.125,
      0.1111111111 1111111111, 0.1, 0.0909090909 0909090909 1]
     ,

     [0.1428571428 5714285714, 0.125, 0.1111111111 1111111111, 0.1,
      0.0909090909 0909090909 1, 0.0833333333 3333333333 3]
     ]
                                                           Type: Matrix Float
                                                   Time: 0.02 (OT) = 0.02 sec
h6F * inverse(h6F)
 

   (15)
   [[1.0,- 0.4 E -14,0.1 E -13,- 0.4 E -13,0.2 E -13,- 0.5 E -14],
    [0.6 E -16,0.9999999999 99999,0.2 E -14,- 0.2 E -13,0.0,- 0.7 E -14],
    [0.6 E -16,- 0.2 E -14,1.0,- 0.2 E -13,0.7 E -14,- 0.4 E -14],
    [0.6 E -16,- 0.2 E -14,0.5 E -14,0.9999999999 9998,0.0,- 0.5 E -14],
    [0.7 E -16,- 0.1 E -14,0.0,- 0.2 E -13,1.0,- 0.4 E -14],
    [0.3 E -16,- 0.9 E -15,0.4 E -14,- 0.1 E -13,0.7 E -14,1.0]]
                                                           Type: Matrix Float
                                       Time: 0.01 (EV) + 0.01 (OT) = 0.02 sec


h6Q := HilbertMatrixQ(6)
 

         +1  1  1  1   1   1 +
         |-  -  -  -   -   - |
         |2  3  4  5   6   7 |
         |                   |
         |1  1  1  1   1   1 |
         |-  -  -  -   -   - |
         |3  4  5  6   7   8 |
         |                   |
         |1  1  1  1   1   1 |
         |-  -  -  -   -   - |
         |4  5  6  7   8   9 |
   (16)  |                   |
         |1  1  1  1   1    1|
         |-  -  -  -   -   --|
         |5  6  7  8   9   10|
         |                   |
         |1  1  1  1    1   1|
         |-  -  -  -   --  --|
         |6  7  8  9   10  11|
         |                   |
         |1  1  1   1   1   1|
         |-  -  -  --  --  --|
         +7  8  9  10  11  12+
                                                Type: Matrix Fraction Integer
                                                                  Time: 0 sec
h6Q * inverse(h6Q)
 

         +1  0  0  0  0  0+
         |                |
         |0  1  0  0  0  0|
         |                |
         |0  0  1  0  0  0|
   (17)  |                |
         |0  0  0  1  0  0|
         |                |
         |0  0  0  0  1  0|
         |                |
         +0  0  0  0  0  1+
                                                Type: Matrix Fraction Integer
                                                   Time: 0.01 (IN) = 0.01 sec


[determinant(HilbertMatrixF(5*n)) for n in 1..10];
 

                                                             Type: List Float
                                       Time: 3.02 (EV) + 0.29 (GC) = 3.31 sec

[determinant(HilbertMatrixQ(5*n)) for n in 1..10];
 

                                                  Type: List Fraction Integer
                                       Time: 2.96 (EV) + 0.38 (GC) = 3.34 sec


[determinant(HilbertMatrixF(5*n)) for n in 1..15];
 

                                                             Type: List Float
                         Time: 0.01 (IN) + 14.09 (EV) + 1.59 (GC) = 15.69 sec

[determinant(HilbertMatrixQ(5*n)) for n in 1..15];
 

                                                  Type: List Fraction Integer
                                     Time: 19.41 (EV) + 2.75 (GC) = 22.16 sec



[determinant(HilbertMatrixF(5*n)) for n in 1..20];
 

                                                             Type: List Float
                                     Time: 42.99 (EV) + 5.07 (GC) = 48.06 sec

[determinant(HilbertMatrixQ(5*n)) for n in 1..20];
 

                                                  Type: List Fraction Integer
                                    Time: 75.55 (EV) + 11.16 (GC) = 86.71 sec


next up previous
Next: Operations on univariate polynomials Up: An introduction to Computer Algebra Previous: Operations on real numbers
Marc Moreno Maza
2003-06-06