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initial svds support based on eigs #9425

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Jan 11, 2015
Merged

initial svds support based on eigs #9425

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9ea6b51
initial svds support based on eigs
jaak-s Dec 20, 2014
8b061df
removed extra spaces, added ritzvec param to svds
jaak-s Dec 21, 2014
d654afa
manual for svds
jaak-s Dec 21, 2014
6e91217
renamed to SVDOperator
jaak-s Dec 21, 2014
95e2d6c
svds now uses eigs of [0 A; A' 0]
jaak-s Jan 2, 2015
b1517a6
support for Complex operators
jaak-s Jan 3, 2015
be1f1bb
removed trailing whitespace
jaak-s Jan 3, 2015
f8edd41
replaced getOperatorType to eltype
jaak-s Jan 3, 2015
b9d7a30
added check for nsv
jaak-s Jan 3, 2015
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1 change: 1 addition & 0 deletions base/exports.jl
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Expand Up @@ -690,6 +690,7 @@ export
svd,
svdfact!,
svdfact,
svds,
svdvals!,
svdvals,
sylvester,
Expand Down
1 change: 1 addition & 0 deletions base/linalg.jl
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Expand Up @@ -102,6 +102,7 @@ export
svd,
svdfact!,
svdfact,
svds,
svdvals!,
svdvals,
sylvester,
Expand Down
48 changes: 48 additions & 0 deletions base/linalg/arnoldi.jl
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Expand Up @@ -106,3 +106,51 @@ function eigs(A, B;
resid, ncv, v, ldv, sigma, iparam, ipntr, workd, workl, lworkl, rwork)

end


## svds

type SVDOperator{T,S} <: AbstractArray{T, 2}
X::S
m::Int
n::Int
SVDOperator(X::S) = new(X, size(X,1), size(X,2))
end

## v = [ left_singular_vector; right_singular_vector ]
*{T,S}(s::SVDOperator{T,S}, v::Vector{T}) = [s.X * v[s.m+1:end]; s.X' * v[1:s.m]]
size(s::SVDOperator) = s.m + s.n, s.m + s.n
issym(s::SVDOperator) = true

function getOperatorType(X)
et = eltype(X)
if et <: Integer || et <: Float64
return Float64
end
if et <: Complex64
return Complex64
end
if et <: Complex
return Complex128
end
if et <: Float32
return Float32
end
error("Element type $et is not supported for 'svds'.")
end

function svds{S}(X::S; nsv::Int = 6, ritzvec::Bool = true, tol::Float64 = 0.0, maxiter::Int = 1000)
otype = getOperatorType(X)
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I think we can just simplify this to use just eltype(X) and avoid getOperatorType() altogether. The checks for whether eltype() is BlasFloat should ideally be in eigs. It should be possible to make SVDOperator free of types then.

ex = eigs(SVDOperator{otype,S}(X), I; ritzvec = ritzvec, nev = 2*nsv, tol = tol, maxiter = maxiter)
ind = [1:2:nsv*2]
sval = abs(ex[1][ind])

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travis doesn't like the trailing whitespace here or on line 151

if ! ritzvec
return sval, ex[2], ex[3], ex[4], ex[5]
end

## calculating singular vectors
left_sv = sqrt(2) * ex[2][ 1:size(X,1), ind ] .* sign(ex[1][ind]')
right_sv = sqrt(2) * ex[2][ size(X,1)+1:end, ind ]
return left_sv, sval, right_sv, ex[3], ex[4], ex[5], ex[6]
end
19 changes: 18 additions & 1 deletion doc/stdlib/linalg.rst
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Expand Up @@ -564,7 +564,7 @@ Linear algebra functions in Julia are largely implemented by calling functions f

Conjugate transpose array ``src`` and store the result in the preallocated array ``dest``, which should have a size corresponding to ``(size(src,2),size(src,1))``. No in-place transposition is supported and unexpected results will happen if `src` and `dest` have overlapping memory regions.

.. function:: eigs(A, [B,]; nev=6, which="LM", tol=0.0, maxiter=1000, sigma=nothing, ritzvec=true, v0=zeros((0,))) -> (d,[v,],nconv,niter,nmult,resid)
.. function:: eigs(A, [B,]; nev=6, which="LM", tol=0.0, maxiter=300, sigma=nothing, ritzvec=true, v0=zeros((0,))) -> (d,[v,],nconv,niter,nmult,resid)

``eigs`` computes eigenvalues ``d`` of ``A`` using Lanczos or Arnoldi iterations for real symmetric or general nonsymmetric matrices respectively. If ``B`` is provided, the generalized eigen-problem is solved. The following keyword arguments are supported:
* ``nev``: Number of eigenvalues
Expand Down Expand Up @@ -600,6 +600,23 @@ Linear algebra functions in Julia are largely implemented by calling functions f
real or complex inverse with level shift ``sigma`` :math:`(A - \sigma I )^{-1}`
=============== ================================== ==================================

.. function:: svds(A; ritzvec=true, args...) -> (left_sv, s, right_sv, nconv, niter, nmult, resid)

``svds`` computes singular values ``s`` of ``A`` using Lanczos or Arnoldi iterations. Uses ``eigs`` underneath so following keyword arguments are supported:
* ``nev``: Number of singular values.
* ``ncv``: Number of Krylov vectors used in the computation; see ``eigs`` manual.
* ``ritzvec``: Whether to return the left and right singular vectors ``left_sv`` and ``right_sv``, default is ``true``. If ``false`` the singular vectors are omitted from the output.
* ``which``: type of singular values (and vectors) to compute, default is largest values. See ``eigs`` manual.
* ``tol``: tolerance, see ``eigs``.
* ``maxiter``: Maximum number of iterations, see ``eigs``.
* ``sigma``: See ``eigs``.
* ``v0``: starting vector of right singular vector from which to start the iterations.

**Example**::

X = sprand(10, 5, 0.2)
svds(X, nev = 2)

.. function:: peakflops(n; parallel=false)

``peakflops`` computes the peak flop rate of the computer by using double precision :func:`Base.LinAlg.BLAS.gemm!`. By default, if no arguments are specified, it multiplies a matrix of size ``n x n``, where ``n = 2000``. If the underlying BLAS is using multiple threads, higher flop rates are realized. The number of BLAS threads can be set with ``blas_set_num_threads(n)``.
Expand Down
58 changes: 58 additions & 0 deletions test/linalg/arnoldi.jl
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@@ -0,0 +1,58 @@
using Base.Test

let # svds test
A = sparse([1, 1, 2, 3, 4], [2, 1, 1, 3, 1], [2.0, -1.0, 6.1, 7.0, 1.5])
S1 = svds(A, nsv = 2)
S2 = svd(full(A))

## singular values match:
@test_approx_eq S1[2] S2[2][1:2]

## 1st left singular vector
s1_left = sign(S1[1][3,1]) * S1[1][:,1]
s2_left = sign(S2[1][3,1]) * S2[1][:,1]
@test_approx_eq s1_left s2_left

## 1st right singular vector
s1_right = sign(S1[3][3,1]) * S1[3][:,1]
s2_right = sign(S2[3][3,1]) * S2[3][:,1]
@test_approx_eq s1_right s2_right
end

let # svd test with boolean matrix
B = [true true false; false true false; true false true; false true true]
S1 = svds(B, nsv = 2)
S2 = svd(B)

## singular values match:
@test_approx_eq S1[2] S2[2][1:2]

## 1st left singular vector
s1_left = sign(S1[1][1,1]) * S1[1][:,1]
s2_left = sign(S2[1][1,1]) * S2[1][:,1]
@test_approx_eq s1_left s2_left

## 1st right singular vector
s1_right = sign(S1[3][1,1]) * S1[3][:,1]
s2_right = sign(S2[3][1,1]) * S2[3][:,1]
@test_approx_eq s1_right s2_right
end

let # complex svds test
A = sparse([1, 1, 2, 3, 4], [2, 1, 1, 3, 1], exp(im*[2.0:2:10]))
S1 = svds(A, nsv = 2)
S2 = svd(full(A))

## singular values match:
@test_approx_eq S1[2] S2[2][1:2]

## left singular vectors
s1_left = abs(S1[1][:,1:2])
s2_left = abs(S2[1][:,1:2])
@test_approx_eq s1_left s2_left

## right singular vectors
s1_right = abs(S1[3][:,1:2])
s2_right = abs(S2[3][:,1:2])
@test_approx_eq s1_right s2_right
end