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Fic. 5.1. An illustration of the preconditioners of Gremban and Miller. A is partitioned hierarchically, and the vertices of B represent subgraphs in that partition. The circled vertex of B represents the subgraph of A consisting of the two circled vertices. The weight of an edge of B is the sum of the weights of the edges of A that connect the subgraph to the rest of A.

Figure 7 Fic. 5.1. An illustration of the preconditioners of Gremban and Miller. A is partitioned hierarchically, and the vertices of B represent subgraphs in that partition. The circled vertex of B represents the subgraph of A consisting of the two circled vertices. The weight of an edge of B is the sum of the weights of the edges of A that connect the subgraph to the rest of A.

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Fic. 2.1. A graph representation of a splitting of A= Ai+---+A4 and B= By +---+B4 such that each A; represents a single edge and each B; is a path that supports the edge A;. This splitting proves that max A7(A, B) < o(A, B) < 4 since the worst congestion-dilation product is 2-2 = 4. the Congestion-Dilation lemma, which they neither state nor prove, to show that each term 7B; — A; is positive semidefinite.
Fic. 2.1. A graph representation of a splitting of A= Ai+---+A4 and B= By +---+B4 such that each A; represents a single edge and each B; is a path that supports the edge A;. This splitting proves that max A7(A, B) < o(A, B) < 4 since the worst congestion-dilation product is 2-2 = 4. the Congestion-Dilation lemma, which they neither state nor prove, to show that each term 7B; — A; is positive semidefinite.
be (k +1)-by-(k +1) matrices with a > 0. Then kB — A is positive semidefinite. Proof. We prove that kB — A is positive semidefinite by showing that the matri
be (k +1)-by-(k +1) matrices with a > 0. Then kB — A is positive semidefinite. Proof. We prove that kB — A is positive semidefinite by showing that the matri
The claim is true before we eliminate row and column 2, since in that case 7 = 1. Assume that the claim is true after we eliminate 7 but before i+1. The elimination of row and column 7+ 1 modifies three entries in C: C11, Citon42, and Crise = Cita. The new values are
The claim is true before we eliminate row and column 2, since in that case 7 = 1. Assume that the claim is true after we eliminate 7 but before i+1. The elimination of row and column 7+ 1 modifies three entries in C: C11, Citon42, and Crise = Cita. The new values are
be a matrix with zero row sums, and with d;,b; > 0 for allt. Then (k-a/ min(b;))B—A is positive semidefinite. Proof. Let b = min(b;). We split B into The matrix B2 is symmetric, diagonally dominant, and has nonpositive off-diagonals, so it is positive semidefinite. We have
be a matrix with zero row sums, and with d;,b; > 0 for allt. Then (k-a/ min(b;))B—A is positive semidefinite. Proof. Let b = min(b;). We split B into The matrix B2 is symmetric, diagonally dominant, and has nonpositive off-diagonals, so it is positive semidefinite. We have
Fic. 4.1. A graph Ga, a spanning tree Gp of Ga (middle), and a spanning tree augmented with extra edges (Gp, right). The augmentation is performed by cutting the tree into subgraphs Vi,...,Va of roughly equal size, and adding the heaviest edge between each pair of subgraphs.
Fic. 4.1. A graph Ga, a spanning tree Gp of Ga (middle), and a spanning tree augmented with extra edges (Gp, right). The augmentation is performed by cutting the tree into subgraphs Vi,...,Va of roughly equal size, and adding the heaviest edge between each pair of subgraphs.
Fic. 6.1. An incomplete-factorization preconditioner for a model problem with zero row sum: (left). The model problem has the same underlying graph, but without the positive dashed edges. The (complete) Cholesky factor of this matrix is the incomplete-Cholesky factor of the model problem. The figure on the right shows the two paths that route each positive edge.
Fic. 6.1. An incomplete-factorization preconditioner for a model problem with zero row sum: (left). The model problem has the same underlying graph, but without the positive dashed edges. The (complete) Cholesky factor of this matrix is the incomplete-Cholesky factor of the model problem. The figure on the right shows the two paths that route each positive edge.
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MathematicsApplied MathematicsComputer ScienceNumerical Analysis and Computational MathematicsCholesky Decomposition
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