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Don't use the macro @. #864

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merged 4 commits into from
May 20, 2024
Merged

Don't use the macro @. #864

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ffdd08c
Don't use the macro @.
amontoison May 20, 2024
9dc980f
Use the correct order of arguments for kcopy!
amontoison May 20, 2024
ffea212
Fix a typo in cg_lanczos_shift.jl
amontoison May 20, 2024
865ee18
fix a typo
amontoison May 20, 2024
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10 changes: 5 additions & 5 deletions src/bilq.jl
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Expand Up @@ -329,19 +329,19 @@ kwargs_bilq = (:c, :transfer_to_bicg, :M, :N, :ldiv, :atol, :rtol, :itmax, :time
# Compute d̅ₖ.
if iter == 1
# d̅₁ = v₁
@. d̅ = vₖ
@kcopy!(n, vₖ, d̅) # d̅ ← vₖ
else
# d̅ₖ = s̄ₖ * d̅ₖ₋₁ - cₖ * vₖ
@kaxpby!(n, -cₖ, vₖ, conj(sₖ), d̅)
end

# Compute vₖ₊₁ and uₖ₊₁.
@. vₖ₋₁ = vₖ # vₖ₋₁ ← vₖ
@. uₖ₋₁ = uₖ # uₖ₋₁ ← uₖ
@kcopy!(n, vₖ, vₖ₋₁) # vₖ₋₁ ← vₖ
@kcopy!(n, uₖ, uₖ₋₁) # uₖ₋₁ ← uₖ

if pᴴq ≠ 0
@. vₖ = q / βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
@. uₖ = p / conj(γₖ₊₁) # γ̄ₖ₊₁uₖ₊₁ = p
vₖ .= q ./ βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
uₖ .= p ./ conj(γₖ₊₁) # γ̄ₖ₊₁uₖ₊₁ = p
end

# Compute ⟨vₖ,vₖ₊₁⟩ and ‖vₖ₊₁‖
Expand Down
16 changes: 8 additions & 8 deletions src/bilqr.jl
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Expand Up @@ -316,7 +316,7 @@ kwargs_bilqr = (:transfer_to_bicg, :atol, :rtol, :itmax, :timemax, :verbose, :hi
# Compute d̅ₖ.
if iter == 1
# d̅₁ = v₁
@. d̅ = vₖ
@kcopy!(n, vₖ, d̅) # d̅ ← vₖ
else
# d̅ₖ = s̄ₖ * d̅ₖ₋₁ - cₖ * vₖ
@kaxpby!(n, -cₖ, vₖ, conj(sₖ), d̅)
Expand Down Expand Up @@ -375,22 +375,22 @@ kwargs_bilqr = (:transfer_to_bicg, :atol, :rtol, :itmax, :timemax, :verbose, :hi
if iter == 2
wₖ₋₁ = wₖ₋₂
@kaxpy!(n, one(FC), uₖ₋₁, wₖ₋₁)
@. wₖ₋₁ = uₖ₋₁ / conj(δₖ₋₁)
wₖ₋₁ .= uₖ₋₁ ./ conj(δₖ₋₁)
end
# w₂ = (u₂ - λ̄₁w₁) / δ̄₂
if iter == 3
wₖ₋₁ = wₖ₋₃
@kaxpy!(n, one(FC), uₖ₋₁, wₖ₋₁)
@kaxpy!(n, -conj(λₖ₋₂), wₖ₋₂, wₖ₋₁)
@. wₖ₋₁ = wₖ₋₁ / conj(δₖ₋₁)
wₖ₋₁ .= wₖ₋₁ ./ conj(δₖ₋₁)
end
# wₖ₋₁ = (uₖ₋₁ - λ̄ₖ₋₂wₖ₋₂ - ϵ̄ₖ₋₃wₖ₋₃) / δ̄ₖ₋₁
if iter ≥ 4
@kscal!(n, -conj(ϵₖ₋₃), wₖ₋₃)
wₖ₋₁ = wₖ₋₃
@kaxpy!(n, one(FC), uₖ₋₁, wₖ₋₁)
@kaxpy!(n, -conj(λₖ₋₂), wₖ₋₂, wₖ₋₁)
@. wₖ₋₁ = wₖ₋₁ / conj(δₖ₋₁)
wₖ₋₁ .= wₖ₋₁ ./ conj(δₖ₋₁)
end

if iter ≥ 3
Expand Down Expand Up @@ -421,12 +421,12 @@ kwargs_bilqr = (:transfer_to_bicg, :atol, :rtol, :itmax, :timemax, :verbose, :hi
end

# Compute vₖ₊₁ and uₖ₊₁.
@. vₖ₋₁ = vₖ # vₖ₋₁ ← vₖ
@. uₖ₋₁ = uₖ # uₖ₋₁ ← uₖ
@kcopy!(n, vₖ, vₖ₋₁) # vₖ₋₁ ← vₖ
@kcopy!(n, uₖ, uₖ₋₁) # uₖ₋₁ ← uₖ

if pᴴq ≠ zero(FC)
@. vₖ = q / βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
@. uₖ = p / conj(γₖ₊₁) # γ̄ₖ₊₁uₖ₊₁ = p
vₖ .= q ./ βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
uₖ .= p ./ conj(γₖ₊₁) # γ̄ₖ₊₁uₖ₊₁ = p
end

# Update ϵₖ₋₃, λₖ₋₂, δbarₖ₋₁, cₖ₋₁, sₖ₋₁, γₖ and βₖ.
Expand Down
4 changes: 2 additions & 2 deletions src/cg_lanczos.jl
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Expand Up @@ -218,9 +218,9 @@ kwargs_cg_lanczos = (:M, :ldiv, :check_curvature, :atol, :rtol, :itmax, :timemax
@kaxpy!(n, -δ, Mv, Mv_next) # Mvₖ₊₁ ← Mvₖ₊₁ - δₖMvₖ
if iter > 0
@kaxpy!(n, -β, Mv_prev, Mv_next) # Mvₖ₊₁ ← Mvₖ₊₁ - βₖMvₖ₋₁
@. Mv_prev = Mv # Mvₖ₋₁ ← Mvₖ
@kcopy!(n, Mv, Mv_prev) # Mvₖ₋₁ ← Mvₖ
end
@. Mv = Mv_next # Mvₖ ← Mvₖ₊₁
@kcopy!(n, Mv_next, Mv) # Mvₖ ← Mvₖ₊₁
MisI || mulorldiv!(v, M, Mv, ldiv) # vₖ₊₁ = M⁻¹ * Mvₖ₊₁
β = sqrt(@kdotr(n, v, Mv)) # βₖ₊₁ = vₖ₊₁ᴴ M vₖ₊₁
@kscal!(n, one(FC) / β, v) # vₖ₊₁ ← vₖ₊₁ / βₖ₊₁
Expand Down
4 changes: 2 additions & 2 deletions src/cg_lanczos_shift.jl
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Expand Up @@ -211,9 +211,9 @@ kwargs_cg_lanczos_shift = (:M, :ldiv, :check_curvature, :atol, :rtol, :itmax, :t
@kaxpy!(n, -δ, Mv, Mv_next) # Mvₖ₊₁ ← Mvₖ₊₁ - δₖMvₖ
if iter > 0
@kaxpy!(n, -β, Mv_prev, Mv_next) # Mvₖ₊₁ ← Mvₖ₊₁ - βₖMvₖ₋₁
@. Mv_prev = Mv # Mvₖ₋₁ ← Mvₖ
@kcopy!(n, Mv, Mv_prev) # Mvₖ₋₁ ← Mvₖ
end
@. Mv = Mv_next # Mvₖ ← Mvₖ₊₁
@kcopy!(n, Mv_next, Mv) # Mvₖ ← Mvₖ₊₁
MisI || mulorldiv!(v, M, Mv, ldiv) # vₖ₊₁ = M⁻¹ * Mvₖ₊₁
β = sqrt(@kdotr(n, v, Mv)) # βₖ₊₁ = vₖ₊₁ᴴ M vₖ₊₁
@kscal!(n, one(FC) / β, v) # vₖ₊₁ ← vₖ₊₁ / βₖ₊₁
Expand Down
4 changes: 2 additions & 2 deletions src/fgmres.jl
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Expand Up @@ -241,7 +241,7 @@ kwargs_fgmres = (:M, :N, :ldiv, :restart, :reorthogonalization, :atol, :rtol, :i
# Initial ζ₁ and V₁
β = @knrm2(n, r₀)
z[1] = β
@. V[1] = r₀ / rNorm
V[1] .= r₀ ./ rNorm

npass = npass + 1
solver.inner_iter = 0
Expand Down Expand Up @@ -332,7 +332,7 @@ kwargs_fgmres = (:M, :N, :ldiv, :restart, :reorthogonalization, :atol, :rtol, :i
push!(V, S(undef, n))
push!(z, zero(FC))
end
@. V[inner_iter+1] = q / Hbis # hₖ₊₁.ₖvₖ₊₁ = q
V[inner_iter+1] .= q ./ Hbis # hₖ₊₁.ₖvₖ₊₁ = q
z[inner_iter+1] = ζₖ₊₁
end
end
Expand Down
4 changes: 2 additions & 2 deletions src/fom.jl
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Expand Up @@ -233,7 +233,7 @@ kwargs_fom = (:M, :N, :ldiv, :restart, :reorthogonalization, :atol, :rtol, :itma
# Initial ζ₁ and V₁
β = @knrm2(n, r₀)
z[1] = β
@. V[1] = r₀ / rNorm
V[1] .= r₀ ./ rNorm

npass = npass + 1
inner_iter = 0
Expand Down Expand Up @@ -314,7 +314,7 @@ kwargs_fom = (:M, :N, :ldiv, :restart, :reorthogonalization, :atol, :rtol, :itma
if !restart && (inner_iter ≥ mem)
push!(V, S(undef, n))
end
@. V[inner_iter+1] = q / Hbis # hₖ₊₁.ₖvₖ₊₁ = q
V[inner_iter+1] .= q ./ Hbis # hₖ₊₁.ₖvₖ₊₁ = q
end
end

Expand Down
4 changes: 2 additions & 2 deletions src/gmres.jl
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Expand Up @@ -235,7 +235,7 @@ kwargs_gmres = (:M, :N, :ldiv, :restart, :reorthogonalization, :atol, :rtol, :it
# Initial ζ₁ and V₁
β = @knrm2(n, r₀)
z[1] = β
@. V[1] = r₀ / rNorm
V[1] .= r₀ ./ rNorm

npass = npass + 1
solver.inner_iter = 0
Expand Down Expand Up @@ -325,7 +325,7 @@ kwargs_gmres = (:M, :N, :ldiv, :restart, :reorthogonalization, :atol, :rtol, :it
push!(V, S(undef, n))
push!(z, zero(FC))
end
@. V[inner_iter+1] = q / Hbis # hₖ₊₁.ₖvₖ₊₁ = q
V[inner_iter+1] .= q ./ Hbis # hₖ₊₁.ₖvₖ₊₁ = q
z[inner_iter+1] = ζₖ₊₁
end
end
Expand Down
8 changes: 4 additions & 4 deletions src/gpmr.jl
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Expand Up @@ -261,12 +261,12 @@ kwargs_gpmr = (:C, :D, :E, :F, :ldiv, :gsp, :λ, :μ, :reorthogonalization, :ato
# βv₁ = Cb
β = @knrm2(m, b₀)
β ≠ 0 || error("b must be nonzero")
@. V[1] = b₀ / β
V[1] .= b₀ ./ β

# γu₁ = Dc
γ = @knrm2(n, c₀)
γ ≠ 0 || error("c must be nonzero")
@. U[1] = c₀ / γ
U[1] .= c₀ ./ γ

# Compute ‖r₀‖² = γ² + β²
rNorm = sqrt(γ^2 + β^2)
Expand Down Expand Up @@ -479,15 +479,15 @@ kwargs_gpmr = (:C, :D, :E, :F, :ldiv, :gsp, :λ, :μ, :reorthogonalization, :ato

# hₖ₊₁.ₖ ≠ 0
if Haux > btol
@. V[k+1] = q / Haux # hₖ₊₁.ₖvₖ₊₁ = q
V[k+1] .= q ./ Haux # hₖ₊₁.ₖvₖ₊₁ = q
else
# Breakdown -- hₖ₊₁.ₖ = ‖q‖₂ = 0 and Auₖ ∈ Span{v₁, ..., vₖ}
V[k+1] .= zero(FC) # vₖ₊₁ = 0 such that vₖ₊₁ ⊥ Span{v₁, ..., vₖ}
end

# fₖ₊₁.ₖ ≠ 0
if Faux > btol
@. U[k+1] = p / Faux # fₖ₊₁.ₖuₖ₊₁ = p
U[k+1] .= p ./ Faux # fₖ₊₁.ₖuₖ₊₁ = p
else
# Breakdown -- fₖ₊₁.ₖ = ‖p‖₂ = 0 and Bvₖ ∈ Span{u₁, ..., uₖ}
U[k+1] .= zero(FC) # uₖ₊₁ = 0 such that uₖ₊₁ ⊥ Span{u₁, ..., uₖ}
Expand Down
4 changes: 2 additions & 2 deletions src/minres.jl
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Expand Up @@ -276,8 +276,8 @@ kwargs_minres = (:M, :ldiv, :λ, :atol, :rtol, :etol, :conlim, :itmax, :timemax,
end
@kaxpy!(n, one(FC) / β, v, w)

@. r1 = r2
@. r2 = y
@kcopy!(n, r2, r1) # r1 ← r2
@kcopy!(n, y, r2) # r2 ← y
MisI || mulorldiv!(v, M, r2, ldiv)
oldβ = β
β = @kdotr(n, r2, v)
Expand Down
4 changes: 2 additions & 2 deletions src/minres_qlp.jl
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Expand Up @@ -369,12 +369,12 @@ kwargs_minres_qlp = (:M, :ldiv, :λ, :atol, :rtol, :Artol, :itmax, :timemax, :ve
# Compute directions wₖ₋₂, ẘₖ₋₁ and w̄ₖ, last columns of Wₖ = Vₖ(Pₖ)ᴴ
if iter == 1
# w̅₁ = v₁
@. wₖ = vₖ
@kcopy!(n, vₖ, wₖ)
elseif iter == 2
# [w̅ₖ₋₁ vₖ] [cpₖ spₖ] = [ẘₖ₋₁ w̅ₖ] ⟷ ẘₖ₋₁ = cpₖ * w̅ₖ₋₁ + spₖ * vₖ
# [spₖ -cpₖ] ⟷ w̅ₖ = spₖ * w̅ₖ₋₁ - cpₖ * vₖ
@kswap(wₖ₋₁, wₖ)
@. wₖ = spₖ * wₖ₋₁ - cpₖ * vₖ
wₖ .= spₖ .* wₖ₋₁ .- cpₖ .* vₖ
@kaxpby!(n, spₖ, vₖ, cpₖ, wₖ₋₁)
else
# [ẘₖ₋₂ w̄ₖ₋₁ vₖ] [cpₖ 0 spₖ] [1 0 0 ] = [wₖ₋₂ ẘₖ₋₁ w̄ₖ] ⟷ wₖ₋₂ = cpₖ * ẘₖ₋₂ + spₖ * vₖ
Expand Down
14 changes: 7 additions & 7 deletions src/qmr.jl
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Expand Up @@ -326,35 +326,35 @@ kwargs_qmr = (:c, :M, :N, :ldiv, :atol, :rtol, :itmax, :timemax, :verbose, :hist
if iter == 1
wₖ = wₖ₋₁
@kaxpy!(n, one(FC), vₖ, wₖ)
@. wₖ = wₖ / δₖ
wₖ .= wₖ ./ δₖ
end
# w₂ = (v₂ - λ₁w₁) / δ₂
if iter == 2
wₖ = wₖ₋₂
@kaxpy!(n, -λₖ₋₁, wₖ₋₁, wₖ)
@kaxpy!(n, one(FC), vₖ, wₖ)
@. wₖ = wₖ / δₖ
wₖ .= wₖ ./ δₖ
end
# wₖ = (vₖ - λₖ₋₁wₖ₋₁ - ϵₖ₋₂wₖ₋₂) / δₖ
if iter ≥ 3
@kscal!(n, -ϵₖ₋₂, wₖ₋₂)
wₖ = wₖ₋₂
@kaxpy!(n, -λₖ₋₁, wₖ₋₁, wₖ)
@kaxpy!(n, one(FC), vₖ, wₖ)
@. wₖ = wₖ / δₖ
wₖ .= wₖ ./ δₖ
end

# Compute solution xₖ.
# xₖ ← xₖ₋₁ + ζₖ * wₖ
@kaxpy!(n, ζₖ, wₖ, x)

# Compute vₖ₊₁ and uₖ₊₁.
@. vₖ₋₁ = vₖ # vₖ₋₁ ← vₖ
@. uₖ₋₁ = uₖ # uₖ₋₁ ← uₖ
@kcopy!(n, vₖ, vₖ₋₁) # vₖ₋₁ ← vₖ
@kcopy!(n, uₖ, uₖ₋₁) # uₖ₋₁ ← uₖ

if pᴴq ≠ zero(FC)
@. vₖ = q / βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
@. uₖ = p / conj(γₖ₊₁) # γ̄ₖ₊₁uₖ₊₁ = p
vₖ .= q ./ βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
uₖ .= p ./ conj(γₖ₊₁) # γ̄ₖ₊₁uₖ₊₁ = p
end

# Compute τₖ₊₁ = τₖ + ‖vₖ₊₁‖²
Expand Down
4 changes: 2 additions & 2 deletions src/symmlq.jl
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Expand Up @@ -303,9 +303,9 @@ kwargs_symmlq = (:M, :ldiv, :transfer_to_cg, :λ, :λest, :atol, :rtol, :etol, :
mul!(Mv_next, A, v)
α = @kdotr(m, v, Mv_next) + λ
@kaxpy!(m, -oldβ, Mvold, Mv_next)
@. Mvold = Mv
@kcopy!(m, Mv, Mvold) # Mvold ← Mv
@kaxpy!(m, -α, Mv, Mv_next)
@. Mv = Mv_next
@kcopy!(m, Mv_next, Mv) # Mv ← Mv_next
MisI || mulorldiv!(v, M, Mv, ldiv)
β = @kdotr(m, v, Mv)
β < 0 && error("Preconditioner is not positive definite")
Expand Down
18 changes: 9 additions & 9 deletions src/tricg.jl
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Expand Up @@ -369,22 +369,22 @@ kwargs_tricg = (:M, :N, :ldiv, :spd, :snd, :flip, :τ, :ν, :atol, :rtol, :itmax
if iter == 1
# [ 1 0 ] [ gx₁ gy₁ ] = [ v₁ 0 ]
# [ δ̄₁ 1 ] [ gx₂ gy₂ ] [ 0 u₁ ]
@. gx₂ₖ₋₁ = vₖ
@. gx₂ₖ = - conj(δₖ) * gx₂ₖ₋₁
@. gy₂ₖ = uₖ
@kcopy!(m, vₖ, gx₂ₖ₋₁) # gx₂ₖ₋₁ ← vₖ
gx₂ₖ .= -conj(δₖ) .* gx₂ₖ₋₁
@kcopy!(n, uₖ, gy₂ₖ) # gy₂ₖ ← uₖ
else
# [ 0 σ̄ₖ 1 0 ] [ gx₂ₖ₋₃ gy₂ₖ₋₃ ] = [ vₖ 0 ]
# [ η̄ₖ λ̄ₖ δ̄ₖ 1 ] [ gx₂ₖ₋₂ gy₂ₖ₋₂ ] [ 0 uₖ ]
# [ gx₂ₖ₋₁ gy₂ₖ₋₁ ]
# [ gx₂ₖ gy₂ₖ ]
@. gx₂ₖ₋₁ = conj(ηₖ) * gx₂ₖ₋₁ + conj(λₖ) * gx₂ₖ
@. gy₂ₖ₋₁ = conj(ηₖ) * gy₂ₖ₋₁ + conj(λₖ) * gy₂ₖ
gx₂ₖ₋₁ .= conj(ηₖ) .* gx₂ₖ₋₁ .+ conj(λₖ) .* gx₂ₖ
gy₂ₖ₋₁ .= conj(ηₖ) .* gy₂ₖ₋₁ .+ conj(λₖ) .* gy₂ₖ

@. gx₂ₖ = vₖ - conj(σₖ) * gx₂ₖ
@. gy₂ₖ = - conj(σₖ) * gy₂ₖ
gx₂ₖ .= vₖ .- conj(σₖ) .* gx₂ₖ
gy₂ₖ .= .- conj(σₖ) .* gy₂ₖ

@. gx₂ₖ₋₁ = - gx₂ₖ₋₁ - conj(δₖ) * gx₂ₖ
@. gy₂ₖ₋₁ = uₖ - gy₂ₖ₋₁ - conj(δₖ) * gy₂ₖ
gx₂ₖ₋₁ .= .- gx₂ₖ₋₁ .- conj(δₖ) .* gx₂ₖ
gy₂ₖ₋₁ .= uₖ .- gy₂ₖ₋₁ .- conj(δₖ) .* gy₂ₖ

# g₂ₖ₋₃ == g₂ₖ and g₂ₖ₋₂ == g₂ₖ₋₁
@kswap(gx₂ₖ₋₁, gx₂ₖ)
Expand Down
16 changes: 8 additions & 8 deletions src/trilqr.jl
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Expand Up @@ -300,7 +300,7 @@ kwargs_trilqr = (:transfer_to_usymcg, :atol, :rtol, :itmax, :timemax, :verbose,
# Compute d̅ₖ.
if iter == 1
# d̅₁ = u₁
@. d̅ = uₖ
@kcopy!(n, uₖ, d̅) # d̅ ← uₖ
else
# d̅ₖ = s̄ₖ * d̅ₖ₋₁ - cₖ * uₖ
@kaxpby!(n, -cₖ, uₖ, conj(sₖ), d̅)
Expand Down Expand Up @@ -351,22 +351,22 @@ kwargs_trilqr = (:transfer_to_usymcg, :atol, :rtol, :itmax, :timemax, :verbose,
if iter == 2
wₖ₋₁ = wₖ₋₂
@kaxpy!(m, one(FC), vₖ₋₁, wₖ₋₁)
@. wₖ₋₁ = vₖ₋₁ / conj(δₖ₋₁)
wₖ₋₁ .= vₖ₋₁ ./ conj(δₖ₋₁)
end
# w₂ = (v₂ - λ̄₁w₁) / δ̄₂
if iter == 3
wₖ₋₁ = wₖ₋₃
@kaxpy!(m, one(FC), vₖ₋₁, wₖ₋₁)
@kaxpy!(m, -conj(λₖ₋₂), wₖ₋₂, wₖ₋₁)
@. wₖ₋₁ = wₖ₋₁ / conj(δₖ₋₁)
wₖ₋₁ .= wₖ₋₁ ./ conj(δₖ₋₁)
end
# wₖ₋₁ = (vₖ₋₁ - λ̄ₖ₋₂wₖ₋₂ - ϵ̄ₖ₋₃wₖ₋₃) / δ̄ₖ₋₁
if iter ≥ 4
@kscal!(m, -conj(ϵₖ₋₃), wₖ₋₃)
wₖ₋₁ = wₖ₋₃
@kaxpy!(m, one(FC), vₖ₋₁, wₖ₋₁)
@kaxpy!(m, -conj(λₖ₋₂), wₖ₋₂, wₖ₋₁)
@. wₖ₋₁ = wₖ₋₁ / conj(δₖ₋₁)
wₖ₋₁ .= wₖ₋₁ ./ conj(δₖ₋₁)
end

if iter ≥ 3
Expand Down Expand Up @@ -399,14 +399,14 @@ kwargs_trilqr = (:transfer_to_usymcg, :atol, :rtol, :itmax, :timemax, :verbose,
end

# Compute uₖ₊₁ and uₖ₊₁.
@. vₖ₋₁ = vₖ # vₖ₋₁ ← vₖ
@. uₖ₋₁ = uₖ # uₖ₋₁ ← uₖ
@kcopy!(m, vₖ, vₖ₋₁) # vₖ₋₁ ← vₖ
@kcopy!(n, uₖ, uₖ₋₁) # uₖ₋₁ ← uₖ

if βₖ₊₁ ≠ zero(T)
@. vₖ = q / βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
vₖ .= q ./ βₖ₊₁ # βₖ₊₁vₖ₊₁ = q
end
if γₖ₊₁ ≠ zero(T)
@. uₖ = p / γₖ₊₁ # γₖ₊₁uₖ₊₁ = p
uₖ .= p ./ γₖ₊₁ # γₖ₊₁uₖ₊₁ = p
end

# Update ϵₖ₋₃, λₖ₋₂, δbarₖ₋₁, cₖ₋₁, sₖ₋₁, γₖ and βₖ.
Expand Down
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