Implement Q-based qrjacobimatrix() in O(n)

.github/workflows/ci.yml

@@ -10,8 +10,6 @@ jobs:
       fail-fast: false
       matrix:
         version:
-          - '1.7'
-          - '1'
           - '^1.9.0-0'
         os:
           - ubuntu-latest

Project.toml

@@ -1,7 +1,7 @@
 name = "ClassicalOrthogonalPolynomials"
 uuid = "b30e2e7b-c4ee-47da-9d5f-2c5c27239acd"
 authors = ["Sheehan Olver <[email protected]>"]
-version = "0.8.1"
+version = "0.9.0"
 
 [deps]
 ArrayLayouts = "4c555306-a7a7-4459-81d9-ec55ddd5c99a"
@@ -43,7 +43,7 @@ LazyArrays = "1.0.1"
 LazyBandedMatrices = "0.8.5"
 QuasiArrays = "0.9.6"
 SpecialFunctions = "1.0, 2"
-julia = "1.7"
+julia = "1.9"
 
 [extras]
 Base64 = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

src/ClassicalOrthogonalPolynomials.jl

@@ -21,7 +21,7 @@ import LazyArrays: MemoryLayout, Applied, ApplyStyle, flatten, _flatten, adjoint
 import ArrayLayouts: MatMulVecAdd, materialize!, _fill_lmul!, sublayout, sub_materialize, lmul!, ldiv!, ldiv, transposelayout, triangulardata,
                         subdiagonaldata, diagonaldata, supdiagonaldata, mul, rowsupport, colsupport
 import LazyBandedMatrices: SymTridiagonal, Bidiagonal, Tridiagonal, unitblocks, BlockRange1, AbstractLazyBandedLayout
-import LinearAlgebra: pinv, factorize, qr, adjoint, transpose, dot, mul!
+import LinearAlgebra: pinv, factorize, qr, adjoint, transpose, dot, mul!, reflectorApply!
 import BandedMatrices: AbstractBandedLayout, AbstractBandedMatrix, _BandedMatrix, bandeddata
 import FillArrays: AbstractFill, getindex_value, SquareEye
 import DomainSets: components

src/choleskyQR.jl

@@ -23,7 +23,7 @@ OrthogonalPolynomial(w::AbstractQuasiVector) = OrthogonalPolynomial(w, orthogona
 function OrthogonalPolynomial(w::AbstractQuasiVector, P::AbstractQuasiMatrix)
     Q = normalized(P)
     X = cholesky_jacobimatrix(w, Q)
-    ConvertedOrthogonalPolynomial(w, X, X.dv.U, Q)
+    ConvertedOrthogonalPolynomial(w, X, parent(X.dv).U, Q)
 end
 
 orthogonalpolynomial(w::AbstractQuasiVector) = OrthogonalPolynomial(w)
@@ -35,7 +35,7 @@ orthogonalpolynomial(w::SubQuasiArray) = orthogonalpolynomial(parent(w))[parenti
 cholesky_jacobimatrix(w, P)
 
 returns the Jacobi matrix `X` associated to a quasi-matrix of polynomials
-orthogonal with respect to `w(x) w_p(x)` where `w_p(x)` is the weight of the polynomials in `P`.
+orthogonal with respect to `w(x) w_p(x)` where `w_p(x)` is the weight of the polynomials in `P` by computing a Cholesky decomposition of the weight modification.
 
 The resulting polynomials are orthonormal on the same domain as `P`. The supplied `P` must be normalized. Accepted inputs are `w` as a function or `W` as an infinite matrix representing multiplication with the function `w` on the basis `P`.
 """
@@ -51,61 +51,62 @@ function cholesky_jacobimatrix(W::AbstractMatrix, Q)
     U = cholesky(W).U
     X = jacobimatrix(Q)
     UX = ApplyArray(*,U,X)
-    return SymTridiagonal(CholeskyJacobiBand{:dv}(U, UX),CholeskyJacobiBand{:ev}(U, UX))
+    CJD = CholeskyJacobiData(U,UX)
+    return SymTridiagonal(view(CJD,:,1),view(CJD,:,2))
 end
 
-# The generated Jacobi operators are symmetric tridiagonal, so we store their data in cached bands
-mutable struct CholeskyJacobiBand{dv,T} <: AbstractCachedVector{T}
-    data::Vector{T}       # store band entries, :dv for diagonal, :ev for off-diagonal
+# The generated Jacobi operators are symmetric tridiagonal, so we store their data in two cached bands which are generated in tandem but can be accessed separately.
+mutable struct CholeskyJacobiData{T} <: AbstractMatrix{T}
+    dv::AbstractVector{T} # store diagonal band entries in adaptively sized vector
+    ev::AbstractVector{T} # store off-diagonal band entries in adaptively sized vector
     U::UpperTriangular{T} # store upper triangular conversion matrix (needed to extend available entries)
     UX::ApplyArray{T}     # store U*X, where X is the Jacobi matrix of the original P (needed to extend available entries)
     datasize::Int         # size of so-far computed block 
 end
 
 # Computes the initial data for the Jacobi operator bands
-function CholeskyJacobiBand{:dv}(U::AbstractMatrix{T}, UX) where T
-    dv = zeros(T,2) # compute a length 2 vector on first go
-    dv[1] = dot(view(UX,1,1), U[1,1] \ [one(T)])
-    dv[2] = dot(view(UX,2,1:2), U[1:2,1:2] \ [zero(T); one(T)])
-    return CholeskyJacobiBand{:dv,T}(dv, U, UX, 2)
-end
-function CholeskyJacobiBand{:ev}(U::AbstractMatrix{T}, UX) where T
-    ev = zeros(T,2) # compute a length 2 vector on first go
-    ev[1] = dot(view(UX,1,1:2), U[1:2,1:2] \ [zero(T); one(T)])
-    ev[2] = dot(view(UX,2,1:3), U[1:3,1:3] \ [zeros(T,2); one(T)])
-    return CholeskyJacobiBand{:ev,T}(ev, U, UX, 2)
+function CholeskyJacobiData(U::AbstractMatrix{T}, UX) where T
+    dv = Vector{T}(undef,2) # compute a length 2 vector on first go
+    ev = Vector{T}(undef,2)
+    dv[1] = UX[1,1]/U[1,1] # this is dot(view(UX,1,1), U[1,1] \ [one(T)])
+    dv[2] = -U[1,2]*UX[2,1]/(U[1,1]*U[2,2])+UX[2,2]/U[2,2] # this is dot(view(UX,2,1:2), U[1:2,1:2] \ [zero(T); one(T)])
+    ev[1] = -UX[1,1]*U[1,2]/(U[1,1]*U[2,2])+UX[1,2]/U[2,2] # this is dot(view(UX,1,1:2), U[1:2,1:2] \ [zero(T); one(T)])
+    ev[2] = UX[2,1]/U[3,3]*(-U[1,3]/U[1,1]+U[1,2]*U[2,3]/(U[1,1]*U[2,2]))+UX[2,2]/U[3,3]*(-U[2,3]/U[2,2])+UX[2,3]/U[3,3] # this is dot(view(UX,2,1:3), U[1:3,1:3] \ [zeros(T,2); one(T)])[1:3,1:3] \ [zeros(T,2); one(T)])
+    return CholeskyJacobiData{T}(dv, ev, U, UX, 2)
 end
 
-size(::CholeskyJacobiBand) = (ℵ₀,) # Stored as an infinite cached vector
+size(::CholeskyJacobiData) = (ℵ₀,2) # Stored as two infinite cached bands
+
+function getindex(K::CholeskyJacobiData, n::Integer, m::Integer)
+    @assert (m==1) || (m==2)
+    resizedata!(K,n,m)
+    m == 1 && return K.dv[n]
+    m == 2 && return K.ev[n]
+end
 
 # Resize and filling functions for cached implementation
-function resizedata!(K::CholeskyJacobiBand, nm::Integer)
+function resizedata!(K::CholeskyJacobiData, n::Integer, m::Integer)
+    nm = max(n,m)
     νμ = K.datasize
     if nm > νμ
-        resize!(K.data,nm)
-        cache_filldata!(K, νμ:nm)
+        resize!(K.dv,nm)
+        resize!(K.ev,nm)
+        _fillcholeskybanddata!(K, νμ:nm)
         K.datasize = nm
     end
     K
 end
-function cache_filldata!(J::CholeskyJacobiBand{:dv,T}, inds::UnitRange{Int}) where T
-    # pre-fill U and UX to prevent expensive step-by-step filling in of cached U and UX in the loop
-    getindex(J.U,inds[end]+1,inds[end]+1)
-    getindex(J.UX,inds[end]+1,inds[end]+1)
 
-    ek = [zero(T); one(T)]
-    @inbounds for k in inds
-        J.data[k] = dot(view(J.UX,k,k-1:k), J.U[k-1:k,k-1:k] \ ek)
-    end
-end
-function cache_filldata!(J::CholeskyJacobiBand{:ev, T}, inds::UnitRange{Int}) where T
+function _fillcholeskybanddata!(J::CholeskyJacobiData{T}, inds::UnitRange{Int}) where T
     # pre-fill U and UX to prevent expensive step-by-step filling in of cached U and UX in the loop
-    getindex(J.U,inds[end]+1,inds[end]+1)
-    getindex(J.UX,inds[end]+1,inds[end]+1)
+    resizedata!(J.U,inds[end]+1,inds[end]+1)
+    resizedata!(J.UX,inds[end]+1,inds[end]+1)
 
-    ek = [zeros(T,2); one(T)]
+    dv, ev, UX, U = J.dv, J.ev, J.UX, J.U
     @inbounds for k in inds
-        J.data[k] = dot(view(J.UX,k,k-1:k+1), J.U[k-1:k+1,k-1:k+1] \ ek)
+        # this is dot(view(UX,k,k-1:k), U[k-1:k,k-1:k] \ ek)
+        dv[k] = -U[k-1,k]*UX[k,k-1]/(U[k-1,k-1]*U[k,k])+UX[k,k]/U[k,k]
+        ev[k] = UX[k,k-1]/U[k+1,k+1]*(-U[k-1,k+1]/U[k-1,k-1]+U[k-1,k]*U[k,k+1]/(U[k-1,k-1]*U[k,k]))+UX[k,k]/U[k+1,k+1]*(-U[k,k+1]/U[k,k])+UX[k,k+1]/U[k+1,k+1]  
     end
 end
 
@@ -114,80 +115,132 @@ end
 qr_jacobimatrix(sqrtw, P)
 
 returns the Jacobi matrix `X` associated to a quasi-matrix of polynomials
-orthogonal with respect to `w(x) w_p(x)` where `w_p(x)` is the weight of the polynomials in `P`.
+orthogonal with respect to `w(x) w_p(x)` where `w_p(x)` is the weight of the polynomials in `P` by computing a QR decomposition of the square root weight modification.
 
 The resulting polynomials are orthonormal on the same domain as `P`. The supplied `P` must be normalized. Accepted inputs for `sqrtw` are the square root of the weight modification as a function or `sqrtW` as an infinite matrix representing multiplication with the function `sqrt(w)` on the basis `P`.
+
+The underlying QR approach allows two methods, one which uses the Q matrix and one which uses the R matrix. To change between methods, an optional argument :Q or :R may be supplied. The default is to use the Q method.
 """
-function qr_jacobimatrix(sqrtw::Function, P)
+function qr_jacobimatrix(sqrtw::Function, P, method = :Q)
     Q = normalized(P)
     x = axes(P,1)
     sqrtW = (Q \ (sqrtw.(x) .* Q))  # Compute weight multiplication via Clenshaw
-    return qr_jacobimatrix(sqrtW, Q)
+    return qr_jacobimatrix(sqrtW, Q, method)
 end
-function qr_jacobimatrix(sqrtW::AbstractMatrix, Q)
+function qr_jacobimatrix(sqrtW::AbstractMatrix{T}, Q, method = :Q) where T
     isnormalized(Q) || error("Polynomials must be orthonormal")
-    SymTridiagonal(QRJacobiBand{:dv}(sqrtW,Q),QRJacobiBand{:ev}(sqrtW,Q))
-end
-
-# The generated Jacobi operators are symmetric tridiagonal, so we store their data in cached bands
-mutable struct QRJacobiBand{dv,T} <: AbstractCachedVector{T}
-    data::Vector{T}       # store band entries, :dv for diagonal, :ev for off-diagonal
-    U::ApplyArray{T}      # store upper triangular conversion matrix (needed to extend available entries)
-    UX::ApplyArray{T}     # store U*X, where X is the Jacobi matrix of the original P (needed to extend available entries)
+    F = qr(sqrtW)
+    QRJD = QRJacobiData{method,T}(F,Q)
+    SymTridiagonal(view(QRJD,:,1),view(QRJD,:,2))
+end
+
+# The generated Jacobi operators are symmetric tridiagonal, so we store their data in two cached bands which are generated in tandem but can be accessed separately.
+mutable struct QRJacobiData{method,T} <: AbstractMatrix{T}
+    dv::AbstractVector{T} # store diagonal band entries in adaptively sized vector
+    ev::AbstractVector{T} # store off-diagonal band entries in adaptively sized vector
+    U                     # store conversion, Q method: stores QR object. R method: only stores R.
+    UX                    # Auxilliary matrix. Q method: stores in-progress incomplete modification. R method: stores U*X for efficiency.
+    P                     # Remember original polynomials
     datasize::Int         # size of so-far computed block 
 end
 
 # Computes the initial data for the Jacobi operator bands
-function QRJacobiBand{:dv}(sqrtW, P::OrthogonalPolynomial{T}) where T
-    U = qr(sqrtW).R
-    U = ApplyArray(*,Diagonal(sign.(view(U,band(0)))),U)
+function QRJacobiData{:Q,T}(F, P) where T
+    b = 3+bandwidths(F.R)[2]÷2
     X = jacobimatrix(P)
-    UX = ApplyArray(*,U,X)
-    dv = zeros(T,2) # compute a length 2 vector on first go
-    dv[1] = dot(view(UX,1,1), U[1,1] \ [one(T)])
-    dv[2] = dot(view(UX,2,1:2), U[1:2,1:2] \ [zero(T); one(T)])
-    return QRJacobiBand{:dv,T}(dv, U, UX, 2)
-end
-function QRJacobiBand{:ev}(sqrtW, P::OrthogonalPolynomial{T}) where T
-    U = qr(sqrtW).R
-    U = ApplyArray(*,Diagonal(sign.(view(U,band(0)))),U)
+        # we fill 1 entry on the first run
+    dv = zeros(T,2)
+    ev = zeros(T,1)
+        # fill first entry (special case)
+    M = Matrix(X[1:b,1:b])
+    resizedata!(F.factors,b,b)
+    # special case for first entry double Householder product
+    v = view(F.factors,1:b,1)
+    reflectorApply!(v, F.τ[1], M)
+    reflectorApply!(v, F.τ[1], M')
+    dv[1] = M[1,1]
+        # fill second entry
+    # computes H*M*H in-place, overwriting M
+    v = view(F.factors,2:b,2)
+    reflectorApply!(v, F.τ[2], view(M,1,2:b))
+    M[1,2:b] .= view(M,1,2:b) # symmetric matrix, avoid recomputation
+    reflectorApply!(v, F.τ[2], view(M,2:b,2:b))
+    reflectorApply!(v, F.τ[2], view(M,2:b,2:b)')
+    ev[1] = M[1,2]*sign(F.R[1,1]*F.R[2,2]) # includes possible correction for sign (only needed in off-diagonal case), since the QR decomposition does not guarantee positive diagonal on R
+    K = Matrix(X[2:b+1,2:b+1])
+    K[1:end-1,1:end-1] .= view(M,2:b,2:b)
+    return QRJacobiData{:Q,T}(dv, ev, F, K, P, 1)
+end
+function QRJacobiData{:R,T}(F, P) where T
+    U = F.R
+    U = ApplyArray(*,Diagonal(sign.(view(U,band(0)))),U)  # QR decomposition does not force positive diagonals on R by default
     X = jacobimatrix(P)
     UX = ApplyArray(*,U,X)
-    ev = zeros(T,2) # compute a length 2 vector on first go
-    ev[1] = dot(view(UX,1,1:2), U[1:2,1:2] \ [zero(T); one(T)])
-    ev[2] = dot(view(UX,2,1:3), U[1:3,1:3] \ [zeros(T,2); one(T)])
-    return QRJacobiBand{:ev,T}(ev, U, UX, 2)
+    dv = Vector{T}(undef,2) # compute a length 2 vector on first go
+    ev = Vector{T}(undef,2)
+    dv[1] = UX[1,1]/U[1,1] # this is dot(view(UX,1,1), U[1,1] \ [one(T)])
+    dv[2] = -U[1,2]*UX[2,1]/(U[1,1]*U[2,2])+UX[2,2]/U[2,2] # this is dot(view(UX,2,1:2), U[1:2,1:2] \ [zero(T); one(T)])
+    ev[1] = -UX[1,1]*U[1,2]/(U[1,1]*U[2,2])+UX[1,2]/U[2,2] # this is dot(view(UX,1,1:2), U[1:2,1:2] \ [zero(T); one(T)])
+    ev[2] = UX[2,1]/U[3,3]*(-U[1,3]/U[1,1]+U[1,2]*U[2,3]/(U[1,1]*U[2,2]))+UX[2,2]/U[3,3]*(-U[2,3]/U[2,2])+UX[2,3]/U[3,3] # this is dot(view(UX,2,1:3), U[1:3,1:3] \ [zeros(T,2); one(T)])
+    return QRJacobiData{:R,T}(dv, ev, U, UX, P, 2)
 end
 
-size(::QRJacobiBand) = (ℵ₀,) # Stored as an infinite cached vector
+
+size(::QRJacobiData) = (ℵ₀,2) # Stored as two infinite cached bands
+
+function getindex(K::QRJacobiData, n::Integer, m::Integer)
+    @assert (m==1) || (m==2)
+    resizedata!(K,n,m)
+    m == 1 && return K.dv[n]
+    m == 2 && return K.ev[n]
+end
 
 # Resize and filling functions for cached implementation
-function resizedata!(K::QRJacobiBand, nm::Integer)
+function resizedata!(K::QRJacobiData, n::Integer, m::Integer)
+    nm = max(n,m)
     νμ = K.datasize
     if nm > νμ
-        resize!(K.data,nm)
-        cache_filldata!(K, νμ:nm)
+        resize!(K.dv,nm)
+        resize!(K.ev,nm)
+        _fillqrbanddata!(K, νμ:nm)
         K.datasize = nm
     end
     K
 end
-function cache_filldata!(J::QRJacobiBand{:dv,T}, inds::UnitRange{Int}) where T
-    # pre-fill U and UX to prevent expensive step-by-step filling in of cached U and UX in the loop
-    getindex(J.U,inds[end]+1,inds[end]+1)
-    getindex(J.UX,inds[end]+1,inds[end]+1)
-
-    ek = [zero(T); one(T)]
-    @inbounds for k in inds
-        J.data[k] = dot(view(J.UX,k,k-1:k), J.U[k-1:k,k-1:k] \ ek)
+function _fillqrbanddata!(J::QRJacobiData{:Q,T}, inds::UnitRange{Int}) where T
+    b = bandwidths(J.U.factors)[2]÷2
+    # pre-fill cached arrays to avoid excessive cost from expansion in loop
+    m, jj = 1+inds[end], inds[2:end]
+    X = jacobimatrix(J.P)[1:m+b+2,1:m+b+2]
+    resizedata!(J.U.factors,m+b,m+b)
+    resizedata!(J.U.τ,m)
+    K, τ, F, dv, ev = J.UX, J.U.τ, J.U.factors, J.dv, J.ev
+    D = sign.(view(J.U.R,band(0)).*view(J.U.R,band(0))[2:end])
+    M = Matrix{T}(undef,b+3,b+3)
+    @inbounds for n in jj
+        dv[n] = K[1,1] # no sign correction needed on diagonal entry due to cancellation
+        # doublehouseholderapply!(K,τ[n+1],view(F,n+2:n+b+2,n+1),w)
+        v = view(F,n+1:n+b+2,n+1)
+        reflectorApply!(v, τ[n+1], view(K,1,2:b+3))
+        M[1,2:b+3] .= view(M,1,2:b+3) # symmetric matrix, avoid recomputation
+        reflectorApply!(v, τ[n+1], view(K,2:b+3,2:b+3))
+        reflectorApply!(v, τ[n+1], view(K,2:b+3,2:b+3)')
+        ev[n] = K[1,2]*D[n] # contains sign correction from QR not forcing positive diagonals
+        M .= view(X,n+1:n+b+3,n+1:n+b+3)
+        M[1:end-1,1:end-1] .= view(K,2:b+3,2:b+3)
+        K .= M
     end
 end
-function cache_filldata!(J::QRJacobiBand{:ev, T}, inds::UnitRange{Int}) where T
+
+function _fillqrbanddata!(J::QRJacobiData{:R,T}, inds::UnitRange{Int}) where T
     # pre-fill U and UX to prevent expensive step-by-step filling in of cached U and UX in the loop
-    getindex(J.U,inds[end]+1,inds[end]+1)
-    getindex(J.UX,inds[end]+1,inds[end]+1)
+    m = inds[end]+1
+    resizedata!(J.U,m,m)
+    resizedata!(J.UX,m,m)
 
-    ek = [zeros(T,2); one(T)]
+    dv, ev, UX, U = J.dv, J.ev, J.UX, J.U
     @inbounds for k in inds
-        J.data[k] = dot(view(J.UX,k,k-1:k+1), J.U[k-1:k+1,k-1:k+1] \ ek)
+        dv[k] = -U[k-1,k]*UX[k,k-1]/(U[k-1,k-1]*U[k,k])+UX[k,k]./U[k,k] # this is dot(view(UX,k,k-1:k), U[k-1:k,k-1:k] \ ek)
+        ev[k] = UX[k,k-1]/U[k+1,k+1]*(-U[k-1,k+1]/U[k-1,k-1]+U[k-1,k]*U[k,k+1]/(U[k-1,k-1]*U[k,k]))+UX[k,k]/U[k+1,k+1]*(-U[k,k+1]/U[k,k])+UX[k,k+1]/U[k+1,k+1]  # this is dot(view(UX,k,k-1:k+1), U[k-1:k+1,k-1:k+1] \ ek)
     end
-end
+end