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Convergence

We now prove convergence of the iterative method for linear equations

  equation929

where tex2html_wrap_inline2036 are linear bounded operators. To avoid purely technical difficulties we assume F and tex2html_wrap_inline1400 to be of finite dimension. The iteration reads

  eqnarray932

We note in passing that for dim tex2html_wrap_inline2042 this is just the Kaczmarz method.

   theorem945

Proof: For tex2html_wrap_inline2048 this is just Theorem 3.9 of [3]. Our proof is a slight extension of this result.

To begin with we write tex2html_wrap_inline1460 in the form

  equation964

with certain vectors tex2html_wrap_inline2052 . Inserting this into the recursion (4.2) yields

displaymath2054

We apply tex2html_wrap_inline1434 . After obvious manipulations we obtain

displaymath2058

Hence,

  equation980

Now let A be the operator tex2html_wrap_inline2062 and R the map tex2html_wrap_inline2066 , i.e. tex2html_wrap_inline2068 . Let tex2html_wrap_inline2070 where D is the (block) diagonal and L the (block) lower left part of A. Let the vectors u, g consist of the components tex2html_wrap_inline1376 , tex2html_wrap_inline1524 , resp. and let tex2html_wrap_inline2086 be the (block) diagonal tex2html_wrap_inline2088 . Then (4.5) reads

displaymath2090

or

  equation993

Using (4.4) with j = p we see that

  equation999

Now assume that tex2html_wrap_inline2094 range tex2html_wrap_inline2096 . Then, tex2html_wrap_inline2098 for all iterates, and (4.6)-(4.7) yield

displaymath2100

or

displaymath2102

Hence

  equation1010

We show that all the eigenvalues of tex2html_wrap_inline2104 are either 1 or less than 1 in absolute value if tex2html_wrap_inline2106 , tex2html_wrap_inline1526 . Let tex2html_wrap_inline2110 be such an eigenvalue and u the corresponding eigenvector, i.e.

displaymath2114

or

displaymath2116

Using tex2html_wrap_inline2118 yields

  equation1023

Now let (u,Du) = 1 and tex2html_wrap_inline2122 , tex2html_wrap_inline2124 with tex2html_wrap_inline2126 , tex2html_wrap_inline2128 , tex2html_wrap_inline2130 real. (4.9) immediately yields

displaymath2132

hence

displaymath2134

Since tex2html_wrap_inline2136 we have tex2html_wrap_inline2138 , and since tex2html_wrap_inline2106 we have tex2html_wrap_inline2142 . For tex2html_wrap_inline2144 we have tex2html_wrap_inline2146 . For tex2html_wrap_inline2148 we have tex2html_wrap_inline2150 provided that tex2html_wrap_inline2152 . Thus tex2html_wrap_inline2154 for tex2html_wrap_inline2148 and tex2html_wrap_inline1440 .

Returning to (4.8) we see that the sequence tex2html_wrap_inline2160 converges except in the eigenspace of tex2html_wrap_inline2104 for the eigenvalue 1. But this eigenspace is identical to the null space of tex2html_wrap_inline2164 . This does not prevent the tex2html_wrap_inline2098 from converging. tex2html_wrap_inline1596


next up previous
Next: References Up: Numerical Solution of Bilinear Previous: Special inverse problems

Frank Wuebbeling
Mon Sep 7 11:40:50 MET DST 1998