MATLAB Mathematics - SERC - Index of
MATLAB Mathematics - SERC - Index of MATLAB Mathematics - SERC - Index of
6 Sparse Matrices 0 10 20 30 40 50 60 0 20 40 60 nz = 420 0 10 20 30 40 50 60 0 20 40 60 nz = 780 0 10 20 30 40 50 60 0 20 40 60 nz = 1380 0 10 20 30 40 50 60 0 20 40 60 nz = 3540 Fill-in is generated by operations like matrix multiplication. The product of two or more matrices usually has more nonzero entries than the individual terms, and so requires more storage. As p increases, B^p fills in and spy(B^p) gets more dense. The distance between two nodes in a graph is the number of steps on the graph necessary to get from one node to the other. The spy plot of the p-th power of B shows the nodes that are a distance p apart. As p increases, it is possible to get to more and more nodes in p steps. For the Bucky ball, B^8 is almost completely full. Only the antidiagonal is zero, indicating that it is possible to get from any node to any other node, except the one directly opposite it on the sphere, in eight steps. 6-22
Adjacency Matrices and Graphs An Airflow Model A calculation performed at NASA’s Research Institute for Applications of Computer Science involves modeling the flow over an airplane wing with two trailing flaps. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 In a two-dimensional model, a triangular grid surrounds a cross section of the wing and flaps. The partial differential equations are nonlinear and involve several unknowns, including hydrodynamic pressure and two components of velocity. Each step of the nonlinear iteration requires the solution of a sparse linear system of equations. Since both the connectivity and the geometric location of the grid points are known, the gplot function can produce the graph shown above. In this example, there are 4253 grid points, each of which is connected to between 3 and 9 others, for a total of 28831 nonzeros in the matrix, and a density equal to 0.0016. This spy plot shows that the node numbering yields a definite band structure. 6-23
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- Page 242 and 243: 6 Sparse Matrices Function Summary
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- Page 246 and 247: 6 Sparse Matrices This matrix requi
- Page 248 and 249: 6 Sparse Matrices S = (3,1) 1 (2,2)
- Page 250 and 251: 6 Sparse Matrices Now F = full(S) d
- Page 252 and 253: 6 Sparse Matrices Importing Sparse
- Page 254 and 255: 6 Sparse Matrices west0479 west0479
- Page 256 and 257: 6 Sparse Matrices The find Function
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- Page 264 and 265: 6 Sparse Matrices 0 500 1000 1500 2
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- Page 268 and 269: 6 Sparse Matrices Similarly, S(:,p)
- Page 270 and 271: 6 Sparse Matrices The following MAT
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- Page 274 and 275: 6 Sparse Matrices QR Factorization
- Page 276 and 277: 6 Sparse Matrices shows that A has
- Page 278 and 279: 6 Sparse Matrices Functions for Ite
- Page 280 and 281: 6 Sparse Matrices set up the five-p
- Page 282 and 283: 6 Sparse Matrices Manipulating Spar
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- Page 286 and 287: Index comparing sparse and full mat
- Page 288 and 289: Index H hb1dae demo 5-35 hb1ode dem
- Page 290 and 291: Index nonstiff ODE examples rigid b
- Page 292 and 293: Index LU factorization 6-30 minimum
- Page 294: Index twobvp demo 5-63 two-dimensio
Adjacency Matrices and Graphs<br />
An Airflow Model<br />
A calculation performed at NASA’s Research Institute for Applications <strong>of</strong><br />
Computer Science involves modeling the flow over an airplane wing with two<br />
trailing flaps.<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
In a two-dimensional model, a triangular grid surrounds a cross section <strong>of</strong> the<br />
wing and flaps. The partial differential equations are nonlinear and involve<br />
several unknowns, including hydrodynamic pressure and two components <strong>of</strong><br />
velocity. Each step <strong>of</strong> the nonlinear iteration requires the solution <strong>of</strong> a sparse<br />
linear system <strong>of</strong> equations. Since both the connectivity and the geometric<br />
location <strong>of</strong> the grid points are known, the gplot function can produce the graph<br />
shown above.<br />
In this example, there are 4253 grid points, each <strong>of</strong> which is connected to<br />
between 3 and 9 others, for a total <strong>of</strong> 28831 nonzeros in the matrix, and a<br />
density equal to 0.0016. This spy plot shows that the node numbering yields a<br />
definite band structure.<br />
6-23
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