Re: SingularValues of symmetric real matrix not converging

*To*: mathgroup at smc.vnet.net*Subject*: [mg25769] Re: [mg25759] SingularValues of symmetric real matrix not converging*From*: "Mark Harder" <harderm at ucs.orst.edu>*Date*: Sun, 22 Oct 2000 23:30:04 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Albert, I hope someone else answers this, too, because I use SingularValues[] all the time, and I can't figure out exactly *why* it won't converge. Obviously, the matrix is a difficult one to work with, since most elements differ only in the 17-th & 18-th decimal places; but I would have thought that evaluating SVD in high precision would handle that: In[151]:= N[SingularValues[mat], 32] SingularValues::"cfail": "Algorithm failed to converge." O.K., so I looked up the error message with the Help facility -- no luck here either; Mathematica with typical grace & aplomb omits this message from its help file. So I tried scaling the matrix by multiplying it by 10^15: SingularValues[1. 10^15 mat] & this returned Transpose [U-matrix], the vector of s-values, and Transpose[V-matrix], but only in the rank-7 approximation. That is because SVD does indeed have a tolerance flag which omits singular values that are too small, and the U- and V- elements that correspond to those omitted s-values. So, to return *all* elements of the SVD of mat, use SingularValues[1. 10^15 mat, Tolerance -> 0 ] , which returns {1.84975`*^14, 1.84975`*^14, 1.84975`*^14, 1.84975`*^14, \ 1.8497499999999997`*^14, 1.8497499999999988`*^14, 1.8497499999999988`*^14, \ 3.1778373115434495`} as the vector of singular values. See the documentation for SingularValues[], which explains how to get the condition number of mat from the above s-vector. To get mat back, we need U.DiagonalMatrix[s].V^T, rescaled by dividing s by 10^15: N[Transpose[utr].DiagonalMatrix[1. 10^-15 sv].vee, 32 ] {{0.161853, -0.0231219, -0.0231219, -0.0231219, -0.0231219, -0.0231219, -0.0 231219, -0.0231219}, {-0.0231219, 0.161853, -0.0231219,.....},