## Group ATandT

Group Description |
Frequency domain analysis of large nonlinear analog circuits, AT&T. See ATT.ps for a paper describing these matrices, in the Other directory: @inproceedings{FeldmannMelvilleLong96, author={Feldmann, P. and Melville, R. and Long, D.}, title={Efficient frequency domain analysis of large nonlinear analog circuits}, booktitle={Proceedings of the IEEE Custom Integrated Circuits Conference}, year={1996} ,address={Santa Clara, CA} ,annote={online, in matrices/ATandT/Documentation/ATT.ps} } -------------------------------------------------------------------------------- rich.m converted to onetone1.rua, and sparse.m converted to onetone2.rua: Date: Wed, 8 May 96 07:30:09 EDT From: David Long <long at research.att.com> To: davis at cise.ufl.edu Subject: matrices I put two matrices in incoming/rich.m.gz and incoming/sparse.m.gz. Both are for the same medium-sized example. The first one has a relatively rich collection of off-diagonal entries and is typical of what we might use for an SSOR-like preconditioner. The second is sparser, but probably more typical of what would be used with direct factorization. Both of these are fairly typical of the matrices that arise in "one-tone" problems, in that the off-diagonal entries are generally confined in some band around the diagonal. I'll also try to find an appropriate matrix from a multi-tone problem. Those tend to have one set of off-diagonal entries close to the diagonal and another set relatively far away. David -------------------------------------------------------------------------------- The tt3.m (multitone) matrix was converted to twotone.rua: Date: Thu, 9 May 96 07:12:46 EDT From: David Long <long at research.att.com> To: davis at cise.ufl.edu Subject: Re: matrices A matrix from a two-tone problem is in incoming/tt3.m.gz. -------------------------------------------------------------------------------- Regarding a question about the paper (ATT.ps): -------------------------------------------------------------------------------- Date: Fri, 10 May 96 14:07:22 EDT From: David Long <long at research.att.com> To: davis at cise.ufl.edu Subject: Re: matrices Yes, figure 2 is supposed to be a diagram of the Jacobian, but let me add a few comments that may help clarify things. First, you can just ignore the Y part. It's included in the formulation for completeness, but is not generally needed. None of the matrices that I sent have a Y part. Second, the matrices can be naturally structured in one of two ways, which we call nN format and Nn format. In nN format, you view the matrix as an n by n block matrix, with each block having size N by N. Similarly, Nn format indicates an N by N block matrix, with each block having size n by n. The two formats are related by the obvious permutations. The picture of the Jacobian in figure 2 is for the matrix in nN format. However, to confuse matters, our program actually builds the preconditioner matrix in Nn format, and that's the format of the matrices that I sent. (Also to confuse matters, the paper is a bit inconsistent. For example, the statement before equation 15 about the Jacobian being block-diagonal for a linear circuit assumes that Jacobian is in Nn format.) The almost-circulant structure in figure 2 is in the dense N by N blocks, i.e., each N by N block is basically a circulant matrix. I say almost-circulant because the multiplication of Gamma C Gamma^-1 by Omega breaks the circulant structure. For simplicity, in the paper, the discussion of the Jacobian structure is carried out assuming that the problem is formulated in the complex domain. In the complex case though, half of the coefficients that define the soluation are just conjugates of the other half. Thus, if you translate everything into the real domain, you can save some memory. The program uses this observation, and that's why the matrices that I sent are real. The Jacobian in the real case is Omega T^-1 Gamma C Gamma^-1 T + T^-1 Gamma G Gamma^-1 T where T is a matrix that translates from the real domain to the complex domain. It turns out that when you apply T^-1 and T to the complex circulant matrix, each pair of conjugate stripes in the circulant maps into one of those tilted boxes that we discussed. You can see these clearly in the tt3.m preconditioner matrix. The paper appeared in the 1996 Custom Integrated Circuits Conference (CICC'96). The conference just ended Wednesday, and I haven't actually seen the proceedings yet, so I can't give you page numbers and such. David -------------------------------------------------------------------------------- My response to the message just above: -------------------------------------------------------------------------------- To: David Long <long at research.att.com> cc: davis at cise.ufl.edu, rcm at allegra.att.com Subject: Re: matrices Date: Fri, 10 May 1996 17:22:19 -0400 From: Tim Davis <davis at cise.ufl.edu> In a previous message, David Long <long at research.att.com> wrote > Yes, figure 2 is supposed to be a diagram of the Jacobian, but let me > add a few comments that may help clarify things... Yes - that clarifies things a lot. I'll have to think it through. In the nN form, the N-by-N blocks are dense? So in the Nn format, the n-by-n blocks have a sparse structure like the left hand side of Figure 2 (the "MNA" matrix)? And the n-by-n matrix has a rather irregular structure, right? The dense blocks, by the way, are why UMFPACK gets good performance on these matrices. You may have observed this ... as you increase N, the mflop rate should increase nicely. The dense blocks are still there, in nN or Nn format, at least in the sense that UMFPACK will find them no matter how you contort the input matrix. In one sense this is an ideal matrix for UMFPACK ... unsymmetric in the n-by-n structure (UMFPACK handles asymmetry and irregular structure fairly well, but there are better methods when the structure is regular), with each element being a dense N-by-N matrix (UMFPACK likes dense blocks, since it gets good performance on them in the dense BLAS kernels). Here are some results so far, on my UltraSparc (128 MB memory, two level cache (Level 1: 16K data and 16K instruction, Level 2: 512K), and 2GB swap space). Peak performance of the BLAS is about 80 mflops in double precision, 160 mflops in single. Theoretical peak is 333 mflops. I've renamed the matrices (rich.m = onetone1, sparse.m = onetone2, and tt3 = twotone). Methods used: UMFPACK2-defaults: with default parameters UMFPACK2-nondef.: no BTF, prefer symmetric pivots UMFPACK2 is the same as MA38 (to appear in Harwell Subr. Library). SuperLU: a sparse partial pivoting code, from Berkeley. Also uses the BLAS. MA48: a successor to MA28. Lots faster than MA28. Does not make as extensive use of the BLAS as does SuperLU and UMFPACK. Better suited to more irregular problems (I've seen it much faster than UMFPACK for some matrices). The following is the factorization time (including ordering, symbolic factorization, etc.): time(cpu) nz in LU flops Mflops in seconds onetone1: UMFPACK2-defaults 60.8 4693767 0.2148D+10 35.3 UMFPACK2-nondef. 61.0 4907410 0.2337D+10 38.2 SuperLU 108.2 4674848 0.2525D+10 23.3 MA48 331.0 5109483 0.4564D+10 13.8 onetone2: UMFPACK2-defaults 10.5 1269790 0.1588D+09 15.1 UMFPACK2-nondef. 9.6 1439817 0.2200D+09 22.9 SuperLU 8.6 1319194 0.1306D+09 15.2 MA48 38.3 1260893 0.2229D+09 5.8 twotone: UMFPACK2-defaults 217.3 9746806 0.6988D+10 32.2 UMFPACK2-nondef. 238.9 15730302 0.9601D+10 40.2 SuperLU failed** MA48 734.1 10860368 1.4675D+10 20.0 For twotone, SuperLU ran out of memory in a preordering part that I wrote, not in SuperLU proper (I need to try it again, and give it more memory in that part). I haven't included pure factorization time - all of these methods can use prior orderings, for example (UMD2FA vs. UMD2RF, for example). Do you use UMD2RF? It would be used if the values of the Jacobian changes, but the pattern remains the same (and the pivot order remains the same, and stays numerical acceptable). In that case, UMFPACK2, SuperLU, and MA48 would all be faster in the 2nd factorization than they would in the first. Do these numbers compare with the results you have? Actually, your BLAS on the RS/6000 should get closer to the theoretical peak than does the UltraSparc. Oh - and thanks for the matrices. Nice to have larger ones in which UMFPACK2 does so nicely. Thanks, Tim -------------------------------------------------------------------------------- Matrix pre2.m, converted to pre2.rua |
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Displaying

**all 4**collection matricesId | Name | Group | Rows | Cols | Nonzeros | Kind | Date | Download File |
---|---|---|---|---|---|---|---|---|

283 | onetone1 | ATandT | 36,057 | 36,057 | 335,552 | Frequency Domain Circuit Simulation Problem | 2001 | MATLAB Rutherford Boeing Matrix Market |

284 | onetone2 | ATandT | 36,057 | 36,057 | 222,596 | Frequency Domain Circuit Simulation Problem | 2001 | MATLAB Rutherford Boeing Matrix Market |

286 | twotone | ATandT | 120,750 | 120,750 | 1,206,265 | Frequency Domain Circuit Simulation Problem | 2001 | MATLAB Rutherford Boeing Matrix Market |

285 | pre2 | ATandT | 659,033 | 659,033 | 5,834,044 | Frequency Domain Circuit Simulation Problem | 2001 | MATLAB Rutherford Boeing Matrix Market |