*James G. Sanderson and Stuart L. Pimm*

- Published in print:
- 2015
- Published Online:
- May 2016
- ISBN:
- 9780226292724
- eISBN:
- 9780226292861
- Item type:
- chapter

- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226292861.003.0005
- Subject:
- Biology, Biodiversity / Conservation Biology

Chapter 5 describes how sample random, or null, matrices are created so that each is representative of the actual, or observed, incidence matrix. Four different methods, or algorithms, to create each ...
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Chapter 5 describes how sample random, or null, matrices are created so that each is representative of the actual, or observed, incidence matrix. Four different methods, or algorithms, to create each null matrix are described: enumeration, swap, construction, and quasi-swap. The quasi-swap method produces a collection of null matrices that is representative of the full null space and samples the full null space uniform-randomLess

Chapter 5 describes how sample random, or null, matrices are created so that each is representative of the actual, or observed, incidence matrix. Four different methods, or algorithms, to create each null matrix are described: enumeration, swap, construction, and quasi-swap. The quasi-swap method produces a collection of null matrices that is representative of the full null space and samples the full null space uniform-random

*Gidon Eshel*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691128917
- eISBN:
- 9781400840632
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691128917.003.0003
- Subject:
- Environmental Science, Environmental Studies

This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with A ɛ ℝM×N, and Gram–Schmidt orthogonalization. In ...
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This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with A ɛ ℝM×N, and Gram–Schmidt orthogonalization. In summary, an M × N matrix is associated with four fundamental spaces. The column space is the set of all M-vectors that are linear combinations of the columns. If the matrix has M independent columns, then the column space is ℝM; otherwise the column space is a subspace of ℝM. Also in ℝM is the left null space, the set of all M-vectors that the matrix’s s transpose maps to the zero N-vector. The row space is the set of all N-vectors that are linear combinations of the rows. If the matrix has N independent rows, then the row space is ℝN; otherwise, the row space is a subspace of ℝN. Also in ℝN is the null space, the set of all N-vectors that the matrix maps to the zero M-vector.Less

This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with **A** ɛ ℝM×N, and Gram–Schmidt orthogonalization. In summary, an *M* × *N* matrix is associated with four fundamental spaces. The *column space* is the set of all *M*-vectors that are linear combinations of the columns. If the matrix has *M* independent columns, then the column space is ℝM; otherwise the column space is a subspace of ℝM. Also in ℝM is the left *null space*, the set of all *M*-vectors that the matrix’s s transpose maps to the zero *N*-vector. The *row space* is the set of all *N*-vectors that are linear combinations of the rows. If the matrix has *N* independent rows, then the row space is ℝN; otherwise, the row space is a subspace of ℝN. Also in ℝN is the *null space*, the set of all *N*-vectors that the matrix maps to the zero *M*-vector.

*Charles L. Epstein and Rafe1 Mazzeo*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0012
- Subject:
- Mathematics, Probability / Statistics

This chapter deals with the semi-group on the space Β⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components ...
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This chapter deals with the semi-group on the space Β⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components before discussing the null-space of the adjoint under the hypothesis that a generalized Kimura diffusion operator, L, meets bP cleanly. It then examines long time asymptotics, along with a lemma in which P is a compact manifold with corners and L is a generalized Kimura diffusion on P. It also considers the existence of irregular solutions to the homogeneous equations Lu = f, for functions that do not belong to the range of the generator of a C⁰-semi-group on Β⁰(P).Less

This chapter deals with the semi-group on the space Β⁰(*P*). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components before discussing the null-space of the adjoint under the hypothesis that a generalized Kimura diffusion operator, *L*, meets *b**P* cleanly. It then examines long time asymptotics, along with a lemma in which *P* is a compact manifold with corners and *L* is a generalized Kimura diffusion on *P*. It also considers the existence of irregular solutions to the homogeneous equations *L**u* = *f*, for functions that do not belong to the range of the generator of a *C*⁰-semi-group on Β⁰(*P*).

*I. S. Duff, A. M. Erisman, and J. K. Reid*

- Published in print:
- 2017
- Published Online:
- April 2017
- ISBN:
- 9780198508380
- eISBN:
- 9780191746420
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508380.003.0014
- Subject:
- Mathematics, Numerical Analysis

We examine the SOLVE phase in the direct solution of sparse systems. Here we assume that the factors have been computed and we study the efficient use of these to determine the solution through ...
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We examine the SOLVE phase in the direct solution of sparse systems. Here we assume that the factors have been computed and we study the efficient use of these to determine the solution through forward and back-substitution. We use the trees described in the previous chapters to study the efficient solution of sparse right-hand sides including the computation of null-space bases. We consider the parallelization of the SOLVE phase.Less

We examine the SOLVE phase in the direct solution of sparse systems. Here we assume that the factors have been computed and we study the efficient use of these to determine the solution through forward and back-substitution. We use the trees described in the previous chapters to study the efficient solution of sparse right-hand sides including the computation of null-space bases. We consider the parallelization of the SOLVE phase.