A modification of the Dewilde–van der Veen method for inversion of finite structured matrices
We study a class of block structured matrices R={ R ij } i, j=1 N with a property that the solution of the corresponding system Rx= y of linear algebraic equations may be performed for O( N) arithmetic operations. In this paper for finite invertible matrices we analyze in detail factorization and in...
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| Vydáno v: | Linear algebra and its applications Ročník 343; s. 419 - 450 |
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| Jazyk: | angličtina |
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01.03.2002
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | We study a class of block structured matrices
R={
R
ij
}
i,
j=1
N
with a property that the solution of the corresponding system
Rx=
y of linear algebraic equations may be performed for O(
N) arithmetic operations. In this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms. These algorithms are related to those suggested by P.M. Dewilde and A.J. van der Veen (Time-varying Systems and Computations, Kluwer Academic Publishers, New York, 1998) for a class of finite and infinite matrices with a small Hankel rank. The algorithms presented here are more transparent and are a modification of the algorithms from the above reference. The approach and the proofs are essentially different from those in the above-mentioned reference. The paper contains also analysis of complexity and results of numerical experiments. |
|---|---|
| AbstractList | We study a class of block structured matrices
R={
R
ij
}
i,
j=1
N
with a property that the solution of the corresponding system
Rx=
y of linear algebraic equations may be performed for O(
N) arithmetic operations. In this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms. These algorithms are related to those suggested by P.M. Dewilde and A.J. van der Veen (Time-varying Systems and Computations, Kluwer Academic Publishers, New York, 1998) for a class of finite and infinite matrices with a small Hankel rank. The algorithms presented here are more transparent and are a modification of the algorithms from the above reference. The approach and the proofs are essentially different from those in the above-mentioned reference. The paper contains also analysis of complexity and results of numerical experiments. |
| Author | Eidelman, Y. Gohberg, I. |
| Author_xml | – sequence: 1 givenname: Y. surname: Eidelman fullname: Eidelman, Y. email: eideyu@math.tau.ac.il – sequence: 2 givenname: I. surname: Gohberg fullname: Gohberg, I. email: gohberg@math.tau.ac.il |
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| Cites_doi | 10.1016/0377-0427(86)90169-X 10.1109/TASSP.1987.1165234 10.1007/BF01225526 10.1007/BF01300581 10.1016/S0898-1221(97)00008-4 10.1090/S0025-5718-1974-0343558-6 10.1002/1099-1506(200101/02)8:1<7::AID-NLA224>3.0.CO;2-Y 10.1007/BF01208381 10.1007/BF01191530 10.1016/S0024-3795(00)00083-5 10.1007/BF01213791 |
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| Keywords | Factorization algorithms Structured matrices Solution of linear equations Inversion algorithms Linear complexity algorithms |
| Language | English |
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| References | Dewilde, van der Veen (BIB2) 2000; 313 Gragg (BIB8) 1986; 16 Gohberg, Kailath, Koltracht (BIB11) 1985; 8 Dewilde, van der Veen (BIB1) 1998 Eidelman, Gohberg (BIB6) 1999; 35 Golub, Van Loan (BIB13) 1983 Gill, Golub, Murray, Saunders (BIB9) 1974; 126 Eidelman, Gohberg (BIB4) 1997; 27 Eidelman, Gohberg (BIB3) 1997; 33 Eidelman, Gohberg (BIB5) 1999; 34 Gohberg, Kailath, Koltracht (BIB12) 1987; 7 Gohberg, Kaashoek (BIB10) 1984; 7 Mastronardi, Chandrasekaran, Van Huffel (BIB14) 2001; 8 Y. Eidelman, I. Gohberg, Fast inversion algorithms for a class block structured matrices, in press Dewilde (10.1016/S0024-3795(01)00363-9_BIB1) 1998 Gill (10.1016/S0024-3795(01)00363-9_BIB9) 1974; 126 Eidelman (10.1016/S0024-3795(01)00363-9_BIB4) 1997; 27 Gragg (10.1016/S0024-3795(01)00363-9_BIB8) 1986; 16 Eidelman (10.1016/S0024-3795(01)00363-9_BIB5) 1999; 34 Eidelman (10.1016/S0024-3795(01)00363-9_BIB6) 1999; 35 10.1016/S0024-3795(01)00363-9_BIB7 Gohberg (10.1016/S0024-3795(01)00363-9_BIB10) 1984; 7 Dewilde (10.1016/S0024-3795(01)00363-9_BIB2) 2000; 313 Gohberg (10.1016/S0024-3795(01)00363-9_BIB12) 1987; 7 Mastronardi (10.1016/S0024-3795(01)00363-9_BIB14) 2001; 8 Golub (10.1016/S0024-3795(01)00363-9_BIB13) 1983 Gohberg (10.1016/S0024-3795(01)00363-9_BIB11) 1985; 8 Eidelman (10.1016/S0024-3795(01)00363-9_BIB3) 1997; 33 |
| References_xml | – volume: 8 start-page: 780 year: 1985 end-page: 804 ident: BIB11 article-title: Linear complexity algorithms for semiseparable matrices publication-title: Integral Equations Operator Theory – reference: Y. Eidelman, I. Gohberg, Fast inversion algorithms for a class block structured matrices, in press – volume: 33 start-page: 69 year: 1997 end-page: 79 ident: BIB3 article-title: Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices publication-title: Comput. Math. Appl. – volume: 126 start-page: 505 year: 1974 end-page: 535 ident: BIB9 article-title: Methods for modifying matrix factorization publication-title: Math. Comp. – volume: 8 start-page: 7 year: 2001 end-page: 12 ident: BIB14 article-title: Fast and stable two-way algorithm for diagonal plus semi-separable systems of linear equations publication-title: Numer. Linear Algebra Appl. – volume: 16 start-page: 1 year: 1986 end-page: 8 ident: BIB8 article-title: The QR algorithm for unitary Hessenberg matrices publication-title: J. Comput. Appl. Math. – volume: 7 start-page: 325 year: 1984 end-page: 391 ident: BIB10 article-title: Time varying linear systems with boundary conditions and integral operators, 1. The transfer operator and its properties publication-title: Integral Equations Operator Theory – volume: 34 start-page: 293 year: 1999 end-page: 324 ident: BIB5 article-title: On a new class of structured matrices publication-title: Integral Equations Operator Theory – volume: 35 start-page: 28 year: 1999 end-page: 52 ident: BIB6 article-title: Linear complexity inversion algorithms for a class of structured matrices publication-title: Integral Equations Operator Theory – year: 1998 ident: BIB1 publication-title: Time-varying Systems and Computations – volume: 7 start-page: 1068 year: 1987 end-page: 1069 ident: BIB12 article-title: A note on diagonal innovation publication-title: Acoustics Speech and Signal Processing – volume: 313 start-page: 53 year: 2000 end-page: 100 ident: BIB2 article-title: Inner–outer factorization and the inversion of locally finite systems of equations publication-title: Linear Algebra Appl. – volume: 27 start-page: 165 year: 1997 end-page: 183 ident: BIB4 article-title: Fast inversion algorithms for diagonal plus semiseparable matrices publication-title: Integral Equations Operator Theory – year: 1983 ident: BIB13 publication-title: Matrix Computations – volume: 16 start-page: 1 year: 1986 ident: 10.1016/S0024-3795(01)00363-9_BIB8 article-title: The QR algorithm for unitary Hessenberg matrices publication-title: J. Comput. Appl. Math. doi: 10.1016/0377-0427(86)90169-X – year: 1998 ident: 10.1016/S0024-3795(01)00363-9_BIB1 – volume: 7 start-page: 1068 year: 1987 ident: 10.1016/S0024-3795(01)00363-9_BIB12 article-title: A note on diagonal innovation publication-title: Acoustics Speech and Signal Processing doi: 10.1109/TASSP.1987.1165234 – volume: 35 start-page: 28 year: 1999 ident: 10.1016/S0024-3795(01)00363-9_BIB6 article-title: Linear complexity inversion algorithms for a class of structured matrices publication-title: Integral Equations Operator Theory doi: 10.1007/BF01225526 – volume: 34 start-page: 293 year: 1999 ident: 10.1016/S0024-3795(01)00363-9_BIB5 article-title: On a new class of structured matrices publication-title: Integral Equations Operator Theory doi: 10.1007/BF01300581 – volume: 33 start-page: 69 year: 1997 ident: 10.1016/S0024-3795(01)00363-9_BIB3 article-title: Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices publication-title: Comput. Math. Appl. doi: 10.1016/S0898-1221(97)00008-4 – volume: 126 start-page: 505 year: 1974 ident: 10.1016/S0024-3795(01)00363-9_BIB9 article-title: Methods for modifying matrix factorization publication-title: Math. Comp. doi: 10.1090/S0025-5718-1974-0343558-6 – volume: 8 start-page: 7 year: 2001 ident: 10.1016/S0024-3795(01)00363-9_BIB14 article-title: Fast and stable two-way algorithm for diagonal plus semi-separable systems of linear equations publication-title: Numer. Linear Algebra Appl. doi: 10.1002/1099-1506(200101/02)8:1<7::AID-NLA224>3.0.CO;2-Y – volume: 7 start-page: 325 year: 1984 ident: 10.1016/S0024-3795(01)00363-9_BIB10 article-title: Time varying linear systems with boundary conditions and integral operators, 1. The transfer operator and its properties publication-title: Integral Equations Operator Theory doi: 10.1007/BF01208381 – volume: 27 start-page: 165 year: 1997 ident: 10.1016/S0024-3795(01)00363-9_BIB4 article-title: Fast inversion algorithms for diagonal plus semiseparable matrices publication-title: Integral Equations Operator Theory doi: 10.1007/BF01191530 – year: 1983 ident: 10.1016/S0024-3795(01)00363-9_BIB13 – volume: 313 start-page: 53 year: 2000 ident: 10.1016/S0024-3795(01)00363-9_BIB2 article-title: Inner–outer factorization and the inversion of locally finite systems of equations publication-title: Linear Algebra Appl. doi: 10.1016/S0024-3795(00)00083-5 – ident: 10.1016/S0024-3795(01)00363-9_BIB7 – volume: 8 start-page: 780 year: 1985 ident: 10.1016/S0024-3795(01)00363-9_BIB11 article-title: Linear complexity algorithms for semiseparable matrices publication-title: Integral Equations Operator Theory doi: 10.1007/BF01213791 |
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| Snippet | We study a class of block structured matrices
R={
R
ij
}
i,
j=1
N
with a property that the solution of the corresponding system
Rx=
y of linear algebraic... |
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| StartPage | 419 |
| SubjectTerms | Factorization algorithms Inversion algorithms Linear complexity algorithms Solution of linear equations Structured matrices |
| Title | A modification of the Dewilde–van der Veen method for inversion of finite structured matrices |
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