Geodesic
Some limit properties of the best determined terms method
Applications of Mathematics, Tome 21 (1976) no. 3, pp. 161-167

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
The properties of the criterion of choice are discussed for the best determined termis method (BDT method). The solution of the problem $Kx=y+\epsilon$, where $K$ is $m\times n$ matrix (ill-conditioned), $x\in R^n, y, \epsilon \in R^m, \sum^m_{i=1} \epsilon^2_i\leq \Delta^2$ and $\Delta 0$ given constant, is rather difficult. The criterion of choice from the set of the vectors $x^{(1)},\ldots, x^{(min(m,n))}$, determined by the BDT method, defines the approximation of the normal solution ok $Kx=y$. This approximation x^{(k)}$ should obey the following properties: $\left\|Kx^{(k)}-(y+\epsilon)\right\|^2\leq \Delta^2$, (ii) if $\left\|Kx^{(j)}-(y+\epsilon)\right\|^2\leq \Delta^2$ the $j\geq k$.
The properties of the criterion of choice are discussed for the best determined termis method (BDT method). The solution of the problem $Kx=y+\epsilon$, where $K$ is $m\times n$ matrix (ill-conditioned), $x\in R^n, y, \epsilon \in R^m, \sum^m_{i=1} \epsilon^2_i\leq \Delta^2$ and $\Delta 0$ given constant, is rather difficult. The criterion of choice from the set of the vectors $x^{(1)},\ldots, x^{(min(m,n))}$, determined by the BDT method, defines the approximation of the normal solution ok $Kx=y$. This approximation x^{(k)}$ should obey the following properties: $\left\|Kx^{(k)}-(y+\epsilon)\right\|^2\leq \Delta^2$, (ii) if $\left\|Kx^{(j)}-(y+\epsilon)\right\|^2\leq \Delta^2$ the $j\geq k$.
DOI : 10.21136/AM.1976.103635
Classification : 45B05, 45L05, 65R05, 65R20 
Neuberg, Jiří. Some limit properties of the best determined terms method. Applications of Mathematics, Tome 21 (1976) no. 3, pp. 161-167. doi: 10.21136/AM.1976.103635
@article{10_21136_AM_1976_103635,
     author = {Neuberg, Ji\v{r}{\'\i}},
     title = {Some limit properties of the best determined terms method},
     journal = {Applications of Mathematics},
     pages = {161--167},
     year = {1976},
     volume = {21},
     number = {3},
     doi = {10.21136/AM.1976.103635},
     mrnumber = {0403272},
     zbl = {0356.45001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1976.103635/}
}
TY  - JOUR
AU  - Neuberg, Jiří
TI  - Some limit properties of the best determined terms method
JO  - Applications of Mathematics
PY  - 1976
SP  - 161
EP  - 167
VL  - 21
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.1976.103635/
DO  - 10.21136/AM.1976.103635
LA  - en
ID  - 10_21136_AM_1976_103635
ER  - 
%0 Journal Article
%A Neuberg, Jiří
%T Some limit properties of the best determined terms method
%J Applications of Mathematics
%D 1976
%P 161-167
%V 21
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1976.103635/
%R 10.21136/AM.1976.103635
%G en
%F 10_21136_AM_1976_103635

[1] G. E. Forsythe С. В. Moler: Computer Solution of Linear Algebraic Systems. Prentice Hall, Englewood Clifs, New Jersey 1967. | MR

[2] R. J. Hanson: A numerical method for solving Fredholm integral equations of the first kind using singular values. SIAM J. Numer. Anal., Vol. 8 (1970), 616-622. | DOI | MR | Zbl

[3] J. M. Varah: On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems. SIAM J. Numer. Anal., Vol. 10 (1973), 257-267. | DOI | MR | Zbl

[4] J. Cifka: The method of the best determined terms. to appear.

[5] J. Hekela: Inverse pomocí metody nejlépe určených termů. to appear in Bull. Astr. Inst. ČSAV.

[6] T. L. Bouillon P. L. Odell: Generalised Inverse Matrices. John Wiley and Sons, London, 1971.

Cité par Sources :