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AdjointIdeal :: backwardSubstitution

backwardSubstitution -- Backward substitution.

Synopsis

Description

Solves the system R*x=b via backward substitution.
i1 : A=random(QQ^3,QQ^3)

o1 = | 1/7 3   2 |
     | 3/4 5/3 4 |
     | 2   3/7 1 |

              3        3
o1 : Matrix QQ  <--- QQ
i2 : (perm,LR)=LRdecomposition(A,j->-j);
i3 : LR

o3 = | 2    3/7       1        |
     | 1/14 291/98    27/14    |
     | 3/8  1771/3492 1027/388 |

              3        3
o3 : Matrix QQ  <--- QQ
i4 : P=transpose (id_(QQ^3))_perm

o4 = | 0 0 1 |
     | 1 0 0 |
     | 0 1 0 |

              3        3
o4 : Matrix QQ  <--- QQ
i5 : R=extractRightUpper(LR)

o5 = | 2 3/7    1        |
     | 0 291/98 27/14    |
     | 0 0      1027/388 |

              3        3
o5 : Matrix QQ  <--- QQ
i6 : L=extractLeftLower(LR)

o6 = | 1    0         0 |
     | 1/14 1         0 |
     | 3/8  1771/3492 1 |

              3        3
o6 : Matrix QQ  <--- QQ
i7 : L*R==P*A

o7 = true
i8 : b=random(QQ^3,QQ^1);

              3        1
o8 : Matrix QQ  <--- QQ
i9 : y=forwardSubstitution(LR,P*b)

o9 = | 9/2       |
     | 5/28      |
     | -971/3492 |

              3        1
o9 : Matrix QQ  <--- QQ
i10 : x=backwardSubstitution(LR,y)

o10 = | 21028/9243 |
      | 791/6162   |
      | -971/9243  |

               3        1
o10 : Matrix QQ  <--- QQ
i11 : inverse(A)*b==x

o11 = true

See also

Ways to use backwardSubstitution :