Groups of banded matrices with banded inverses

Strang, Gilbert

doi:10.1090/s0002-9939-2011-10959-6

Cited by 10 publications

(10 citation statements)

References 9 publications

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“…This section describes how to find an appropriate fairing operation from Strang's groups of banded matrices with banded inverses [78].…”

Section: Methodsmentioning

confidence: 99%

“…In a banded matrix, non-zero entries are in a band w along the diagonal and a ij = 0 for |i − j| > w. Strang [77] proves that a banded matrix has a banded inverse if it can be factorized into a product of block diagonal matrices with invertible blocks. Using this property he establishes a group of banded matrices with banded inverses [78].…”

Section: Chasing Gamementioning

confidence: 99%

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Smooth reverse subdivision

Sadeghi

Samavati

2009

Computers & Graphics

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Multiresolution (MR) provides a useful framework for hierarchical representation and manipulation of geometric objects. It consists of two main operations: decomposition (finding coarse points and details) and reconstruction (subdivision plus error correction). One way to construct MR is to use a reverse subdivision (RS) approach by minimizing the error between fine points and subdivided coarse points. However, after a few levels of decomposition using current RS techniques, the coarse points are usually high energy and do not preserve the overall structure of the fine points. This limits the reverse subdivision-based editing and synthesis applications to produce accurate results. This thesis introduces a new reverse subdivision framework, entitled "Smooth Reverse Subdivision", that considers the smoothness of the coarse points as a factor in the decomposition. Using a weighted least-squares approach, a trial set of reverse subdivision xi 5.20 Two applications of the new fairing operation on a coarse mesh.. .. .. 5.21 Mesh editing using the new local fairing operation. (a) Fine mesh. (b) Coarse mesh. (c) Smooth coarse mesh. (d) Edited smooth coarse mesh.

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“…This section describes how to find an appropriate fairing operation from Strang's groups of banded matrices with banded inverses [78].…”

Section: Methodsmentioning

confidence: 99%

Section: Chasing Gamementioning

confidence: 99%

Smooth reverse subdivision

Sadeghi

Samavati

2009

Computers & Graphics

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“…This short section establishes the factorizations P c = BC and Both factorizations are particularly simple cases of a more general theorem [22,23] for banded matrices with banded inverses. For permutations, P and P −1 = P ⊤ have the same bandwidth w. Figure 4.1 below shows a typical matrix P c with w = 2.…”

Section: Factorizations Of Banded Permutationsmentioning

confidence: 99%

The main diagonal of a permutation matrix

Lindner¹,

Strang²

2013

Linear Algebra and its Applications

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By counting 1's in the "right half" of 2w consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth w. Then the matrix can be correctly centered and factored into blockdiagonal permutation matrices.Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is determined "at infinity" in general, but from only 2w rows for banded permutations.Mathematics subject classification (2010): 15A23; 47A53, 47B36.

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“…Suppose q > 0. By observation (5), k may be moved to the right side by q greedy steps, in which k is most difficult for k 0 to overtake. We may assume that this worst case happens, and k is moved to the right by q steps before it is overtaken by k 0 , so k will be at the i l þ q position.…”

mentioning

confidence: 99%

“…Assume that there are r elements on the left of k that are also larger than k 0 , and among them, r 1 elements are on the left of k 0 so that i j − r 1 ≥ u 1 . Then by observation (5), k 0 may be moved to the left by r 1 greedy steps before k 0 overtakes k. After that, all r elements may force k 0 to remain stationary for r þ 1 greedy steps, by observation (6).…”

mentioning

confidence: 99%