Minimal iteration method

A method for solving linear algebraic equations $ A x = b $, in which the solution $ x $ is represented as a linear combination of basis vectors which are orthogonal in some metric connected with the matrix of the system.

In the case of a symmetric matrix $ A $, the orthogonal system of vectors $ p _ {0} \dots p _ {n-1} $ is constructed using the three-term recurrence formula

$$ \tag{1 } p _ {k+1} = A p _ {k} - \alpha _ {k} p _ {k} - \beta _ {k} p _ {k-1} ,\ \ k = 1 \dots n - 2 , $$

$ p _ {1} = A p _ {0} - \alpha _ {0} p _ {0} $, $ p _ {0} $ an arbitrary vector, where

$$ \alpha _ {k} = \ \frac{( A p _ {k} , p _ {k} ) }{( p _ {k} , p _ {k} ) } ,\ \ k = 0 \dots n - 2 , $$

$$ \beta _ {k} = \frac{( p _ {k} , p _ {k} ) }{( p _ {k-1} , p _ {k-1} ) } ,\ k = 1 \dots n - 2 . $$

The solution of the system $ A x = b $ is found by the formula $ x = \sum_{k=0}^ {n-1} c _ {k} p _ {k} $, and the coefficients $ c _ {k} $ are given as the solutions of the system

$$ \tag{2 } \left . \begin{array}{c} c _ {i-1} + \alpha _ {i} c _ {i} + \beta _ {i+1} c _ {i+1} = \ \frac{( b , p _ {i} ) }{( p _ {i} , p _ {i} ) } ,\ \ i = 1 \dots n - 2 , \\ \alpha _ {0} c _ {0} + \beta _ {1} c _ {1} = \ \frac{( b , p _ {0} ) }{( p _ {0} , p _ {0} ) } , \\ c _ {n-2} + \alpha _ {n-1} c _ {n-1} = \ \frac{( b , p _ {n-1} ) }{( p _ {n-1} , p _ {n-1} ) } . \end{array} \right \} $$

If the orthogonalization algorithm is degenerate, that is, if $ p _ {r} = 0 $ for $ r < n $, one has to choose a new initial vector $ p _ {0} ^ {(1)} $, orthogonal to $ p _ {0} \dots p _ {r-1} $ and one has to complete the system of basis vectors to a complete system.

In the case of a non-symmetric matrix a bi-orthogonal algorithm is used.

If $ A $ is symmetric and positive definite, then constructing an $ A $- orthogonal system $ p _ {0} \dots p _ {n-1} $ by formula (1) with

$$ \alpha _ {k} = \ \frac{( A {p _ {k} } , A {p _ {k} } ) }{( A {p _ {k} } , p _ {k} ) } ,\ \ \beta _ {k} = \ \frac{( A {p _ {k} } , p _ {k} ) }{( A {p _ {k-1} } , p _ {k-1} ) } $$

enables one to avoid solving the auxiliary system (2) and gives an explicit expression for the coefficients $ c _ {k} $: $ c _ {k} = ( b , p _ {k} ) / ( A p _ {k} , p _ {k} ) $. Here, to the method of $ A $- minimal iteration one can add the iteration

$$ x _ {k+1} = x _ {k} + c _ {k+1} p _ {k+1} ,\ \ k = 0 \dots n - 2 ,\ \ x _ {0} = c _ {0} p _ {0} , $$

where $ x = x _ {n-1} $. This modification of the method does not require a repeated use of all the vectors $ p _ {0} \dots p _ {k-1} $. A minimal iteration method is used also for the solution of the complete eigen value problem and for finding the inverse matrix.

References

[1]	C. Lanczos, "An iteration method for the solution of the eigenvalue problem of linear differential and integral operators" Res. Nat. Bur. Stand. , 45 : 4 (1950) pp. 255–288
[2]	D.K. Faddeev, V.N. Faddeeva, "Computational methods of linear algebra" , Freeman (1963) (Translated from Russian)