Matrix: JGD_Trefethen/Trefethen_20000b

Description: Diagonal matrices with primes, Nick Trefethen, Oxford Univ.

 (undirected graph drawing)

• Matrix group: JGD_Trefethen
• download as a MATLAB mat-file, file size: 1018 KB. Use UFget(2212) or UFget('JGD_Trefethen/Trefethen_20000b') in MATLAB.

 Matrix properties number of rows 19,999 number of columns 19,999 nonzeros 554,435 structural full rank? yes structural rank 19,999 # of blocks from dmperm 1 # strongly connected comp. 1 explicit zero entries 0 nonzero pattern symmetry symmetric numeric value symmetry symmetric type integer structure symmetric Cholesky candidate? yes positive definite? yes

 author N. Trefethen editor J.-G. Dumas date 2008 kind combinatorial problem 2D/3D problem? no

Notes:

```Diagonal matrices with primes, Nick Trefethen, Oxford Univ.
From Jean-Guillaume Dumas' Sparse Integer Matrix Collection,
http://ljk.imag.fr/membres/Jean-Guillaume.Dumas/simc.html

Problem 7 of the Hundred-dollar, Hundred-digit Challenge Problems,
SIAM News, vol 35, no. 1.

7. Let A be the 20,000 x 20,000 matrix whose entries are zero
everywhere except for the primes 2, 3, 5, 7, . . . , 224737 along the
main diagonal and the number 1 in all the positions A(i,j) with
|i-j| = 1,2,4,8, . . . ,16384.  What is the (1,1) entry of inv(A)?

http://www.siam.org/news/news.php?id=388

Filename in JGD collection: Trefethen/trefethen_20000__19999_minor.sms
```

 Ordering statistics: result nnz(chol(P*(A+A'+s*I)*P')) with AMD 82,368,615 Cholesky flop count 6.9e+11 nnz(L+U), no partial pivoting, with AMD 164,717,231 nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD 173,742,045 nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD 188,532,302

 SVD-based statistics: norm(A) 224737 min(svd(A)) 2.34328 cond(A) 95906.9 rank(A) 19,999 sprank(A)-rank(A) 0 null space dimension 0 full numerical rank? yes

 singular values (MAT file): click here SVD method used: s = svd (full (R)) ; where [~,R,E] = spqr (A) with droptol of zero status: ok