## Dept. of Mathematics and Computer Science (Berlin)

**reviewer:** Miloslav Znojil
**reviewernum:** 9689

**revieweremail:**
znojil@ujf.cas.cz

**zblno:** DE015197637

**author:**
Golub, Gene H. et al

**shorttitle:** A stable numerical
method....

**source:** SIAM J. Sci. Comput. 21, No. 4,
1222-1243 (2000)

**rpclass:** 65F15

**rsclass:** 65D32
44A60 65F35 65E05 94A12 86A20

**keywords:**shape ..
inversion .. moments .. matrix pencils .. polygon ..
ill-conditioned .. quaddrature .. gravimetry

**revtext:**
Reconstruction of a polygon in the complex plane from its moments
is studied. The paper presents a stable method. In its outline (a
Hankel-matrix generalized eigenvalue problem), the key points are
the clarification of the sensitivity (with numerical
illustrations) and improved conditioning (via shifts of moments
and their diagonal scaling). Motivation is beyond any doubt:
Similar problems are currently very ill conditioned though often
needed in practice. In order to underline that, an explicit
application is offered at the end (in a geophysical reconstruction
of an anomalous domain from its gravimetric measurements) and
several others are listed (notably: tomography). In a historical
comment a duality of the problem in question to a 2-D numerical
quadrature (in a way generalizing the Motzkin and Schoenberg
formula for triangular regions) is pointed out, and a remark is
added recalling its connection (and a broader grasp in comparison)
with the matrix pencil solution of certain signal decomposition
problems.

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