improve ldiv method, add LAPACK license text

Author: simonbyrne

LICENSE.md

@@ -20,3 +20,37 @@ The GenericSVD.jl package is licensed under the MIT "Expat" License:
 > CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
 > TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
 > SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+`svdvals2x2` is a Julia translation of the LAPACK DLAS2 function, which is made available under the LAPACK license:
+
+> Copyright (c) 1992-2008 The University of Tennessee.  All rights reserved.
+>
+> Additional copyrights may follow
+>
+> Redistribution and use in source and binary forms, with or without
+> modification, are permitted provided that the following conditions are
+> met:
+>
+> - Redistributions of source code must retain the above copyright
+>   notice, this list of conditions and the following disclaimer.
+>
+> - Redistributions in binary form must reproduce the above copyright
+>   notice, this list of conditions and the following disclaimer listed
+>   in this license in the documentation and/or other materials
+>   provided with the distribution.
+>
+> - Neither the name of the copyright holders nor the names of its
+>   contributors may be used to endorse or promote products derived from
+>   this software without specific prior written permission.
+>
+> THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+> "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+> LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+> A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+> OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+> SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+> LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+> DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+> THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+> (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+> OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

src/utils.jl

@@ -27,7 +27,6 @@ A_mul_B!(G::LinAlg.Givens, ::Void) = nothing
 A_mul_Bc!(::Void, G::LinAlg.Givens) = nothing
 
 function A_ldiv_B!{Ta,Tb}(A::SVD{Ta}, B::StridedVecOrMat{Tb})
-    T = promote_type(Ta,Tb)
-    k = length(find(A.S .> eps(real(T))*maximum(A.S)))
+    k = searchsortedlast(A.S, eps(real(Ta))*A.S[1], rev=true)
     A.Vt[1:k,:]' * (A.S[1:k] .\ (A.U[:,1:k]' * B))
 end

test/runtests.jl

@@ -22,7 +22,7 @@ bS = svdfact(bX)
 
 
 bY = big(randn(n))
-@test qrfact(bX,Val{false}) \ bY ≈ bS \ bY
+@test isapprox(qrfact(bX,Val{false}) \ bY, bS \ bY, rtol=1e3*eps(BigFloat))
 
 X = randn(n,m)+im*randn(n,m)
 bX = big(X)