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DiffAlg :: linearComb

linearComb -- generic linear combination of elements

Synopsis

Description

This routine produce a generic linear combination of the elements in L. It can be used together with genKer or genIm to solve a system of homogeneous linear equations.


In this example we compute a generic and a particular linear combination of two particular differential 2-forms.

i1 : w = random newForm(2,1,2,"a")

        2             2                            2            2                   2            2             2                
o1 = (2x  - 2x x  + 6x  - 3x x  + 6x x )dx  + (- 7x  + 5x x  - x  - 2x x  - x x  + x )dx  + (- 3x  - 4x x  + 7x  + 2x x  - 7x x 
        0     0 1     1     0 2     1 2   0        0     0 1    1     0 2    1 2    2   1        0     0 1     1     0 2     1 2
     ----------------------------------------------------------------------------------------------------------------------------
         2
     - 3x )dx
         2   2

o1 : DiffAlgForm
i2 : h = random newForm(2,1,2,"a")

       2             2                    2           2     2                   2          2             2            2
o2 = (x  + 7x x  + 3x  + 2x x  + x x  + 8x )dx  + (- x  - 4x  - 7x x  - x x  - x )dx  + (3x  - 4x x  + 7x  - 5x x  + x )dx
       0     0 1     1     0 2    1 2     2   0       0     1     0 2    1 2    2   1      0     0 1     1     0 2    2   2

o2 : DiffAlgForm
i3 : linearComb({w,h},"a")

                 2                                   2                                            2                     2  
o3 = ((2a  + a )x  + (- 2a  + 7a )x x  + (6a  + 3a )x  + (- 3a  + 2a )x x  + (6a  + a )x x  + 8a x )dx  + ((- 7a  - a )x  +
         0    1  0        0     1  0 1      0     1  1        0     1  0 2      0    1  1 2     1 2   0         0    1  0  
     ----------------------------------------------------------------------------------------------------------------------------
                            2                                                   2                      2                      
     5a x x  + (- a  - 4a )x  + (- 2a  - 7a )x x  + (- a  - a )x x  + (a  - a )x )dx  + ((- 3a  + 3a )x  + (- 4a  - 4a )x x  +
       0 0 1       0     1  1        0     1  0 2       0    1  1 2     0    1  2   1         0     1  0        0     1  0 1  
     ----------------------------------------------------------------------------------------------------------------------------
                 2                                            2
     (7a  + 7a )x  + (2a  - 5a )x x  - 7a x x  + (- 3a  + a )x )dx
        0     1  1      0     1  0 2     0 1 2        0    1  2   2

o3 : DiffAlgForm
i4 : random oo

        2               2                       2             2               2                                       2          
o4 = (6x  + 10x x  + 18x  - 2x x  + 14x x  + 16x )dx  + (- 16x  + 10x x  - 10x  - 18x x  - 4x x )dx  + (- 16x x  + 28x  - 6x x  -
        0      0 1      1     0 2      1 2      2   0         0      0 1      1      0 2     1 2   1         0 1      1     0 2  
     ----------------------------------------------------------------------------------------------------------------------------
                2
     14x x  - 4x )dx
        1 2     2   2

o4 : DiffAlgForm

In this example we compute a generic differential 1-form that descends to the projective plane. Then, we impose another linear condition.

i5 : w = newForm(2,1,2,"a");
i6 : h = random newForm(2,2,1,"a");
i7 : L = genKer( (radial 2) _ w,w)

                 2                2                         2                                             2      2              
o7 = {x x dx  - x dx , x x dx  - x dx , x x dx  - x x dx , x dx  - x x dx , x x dx  - x x dx , x x dx  - x dx , x dx  - x x dx ,
       0 1  0    0  1   0 2  0    0  2   0 2  1    0 1  2   1  0    0 1  1   1 2  0    0 1  2   1 2  1    1  2   2  0    0 2  2 
     ----------------------------------------------------------------------------------------------------------------------------
      2
     x dx  - x x dx }
      2  1    1 2  2

o7 : List
i8 : wr = linearComb(L,"a")

                  2                        2             2                                 2             2                    
o8 = (a x x  + a x  + a x x  + a x x  + a x )dx  + (- a x  - a x x  + a x x  + a x x  + a x )dx  + (- a x  + (- a  - a )x x  -
       0 0 1    3 1    1 0 2    4 1 2    6 2   0       0 0    3 0 1    2 0 2    5 1 2    7 2   1       1 0       2    4  0 1  
     ----------------------------------------------------------------------------------------------------------------------------
        2
     a x  - a x x  - a x x )dx
      5 1    6 0 2    7 1 2   2

o8 : DiffAlgForm
i9 : genKer(h ^ wr, wr)

       5 2   5       3       3 2          5       3       2        2          5 2   6       2 2   3
o9 = {(-x  + -x x  + -x x  + -x )dx  + (- -x x  + -x x  + -x x  + x )dx  + (- -x  - -x x  - -x  - -x x  - x x )dx }
       7 1   7 0 2   7 1 2   7 2   0      7 0 1   7 0 2   7 1 2    2   1      7 0   7 0 1   7 1   7 0 2    1 2   2

o9 : List

See also

Ways to use linearComb :

For the programmer

The object linearComb is a method function.