Add a function knorm_elliptic

JuliaSmoothOptimizers · Oct 17, 2024 · 63c0806 · 63c0806
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diff --git a/src/car.jl b/src/car.jl
@@ -157,7 +157,7 @@ kwargs_car = (:M, :ldiv, :atol, :rtol, :itmax, :timemax, :verbose, :history, :ca
     history && push!(rNorms, rNorm)
 
     # Compute ‖Ar₀‖
-    ArNorm = MisI ? knorm(n, s) : sqrt(kdotr(n, r, u))
+    ArNorm = MisI ? knorm(n, s) : knorm_elliptic(n, r, u)
     history && push!(ArNorms, ArNorm)
 
     if rNorm == 0
@@ -210,7 +210,7 @@ kwargs_car = (:M, :ldiv, :atol, :rtol, :itmax, :timemax, :verbose, :history, :ca
         kaxpby!(n, one(FC), t, β, u)  # uₖ₊₁ = tₖ₊₁ + βₖ * uₖ
 
         # Compute ‖Arₖ‖
-        ArNorm = MisI ? knorm(n, s) : sqrt(kdotr(n, r, u))
+        ArNorm = MisI ? knorm(n, s) : knorm_elliptic(n, r, u)
         history && push!(ArNorms, ArNorm)
       end
 

diff --git a/src/cg_lanczos.jl b/src/cg_lanczos.jl
@@ -138,7 +138,7 @@ kwargs_cg_lanczos = (:M, :ldiv, :check_curvature, :atol, :rtol, :itmax, :timemax
       kcopy!(n, Mv, b)  # Mv ← b
     end
     MisI || mulorldiv!(v, M, Mv, ldiv)  # v₁ = M⁻¹r₀
-    β = sqrt(kdotr(n, v, Mv))           # β₁ = v₁ᴴ M v₁
+    β = knorm_elliptic(n, v, Mv)        # β₁ = v₁ᴴ M v₁
     σ = β
     rNorm = σ
     history && push!(rNorms, rNorm)
@@ -201,7 +201,7 @@ kwargs_cg_lanczos = (:M, :ldiv, :check_curvature, :atol, :rtol, :itmax, :timemax
       end
       kcopy!(n, Mv, Mv_next)              # 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
+      β = knorm_elliptic(n, v, Mv)        # βₖ₊₁ = vₖ₊₁ᴴ M vₖ₊₁
       kscal!(n, one(FC) / β, v)           # vₖ₊₁  ←  vₖ₊₁ / βₖ₊₁
       MisI || kscal!(n, one(FC) / β, Mv)  # Mvₖ₊₁ ← Mvₖ₊₁ / βₖ₊₁
       Anorm2 += β_prev^2 + β^2 + δ^2      # Use ‖Tₖ₊₁‖₂ as increasing approximation of ‖A‖₂.

diff --git a/src/cg_lanczos_shift.jl b/src/cg_lanczos_shift.jl
@@ -133,7 +133,7 @@ kwargs_cg_lanczos_shift = (:M, :ldiv, :check_curvature, :atol, :rtol, :itmax, :t
     end
     kcopy!(n, Mv, b)                    # Mv₁ ← b
     MisI || mulorldiv!(v, M, Mv, ldiv)  # v₁ = M⁻¹ * Mv₁
-    β = sqrt(kdotr(n, v, Mv))           # β₁ = v₁ᴴ M v₁
+    β = knorm_elliptic(n, v, Mv)        # β₁ = v₁ᴴ M v₁
     kfill!(rNorms, β)
     if history
       for i = 1 : nshifts
@@ -205,7 +205,7 @@ kwargs_cg_lanczos_shift = (:M, :ldiv, :check_curvature, :atol, :rtol, :itmax, :t
       end
       kcopy!(n, Mv, Mv_next)              # 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
+      β = knorm_elliptic(n, v, Mv)        # βₖ₊₁ = vₖ₊₁ᴴ M vₖ₊₁
       kscal!(n, one(FC) / β, v)           # vₖ₊₁  ←  vₖ₊₁ / βₖ₊₁
       MisI || kscal!(n, one(FC) / β, Mv)  # Mvₖ₊₁ ← Mvₖ₊₁ / βₖ₊₁
 

diff --git a/src/cgls_lanczos_shift.jl b/src/cgls_lanczos_shift.jl
@@ -137,10 +137,10 @@ kwargs_cgls_lanczos_shift = (:M, :ldiv, :atol, :rtol, :itmax, :timemax, :verbose
       kfill!(x[i], zero(FC))  # x₀
     end
 
-    kcopy!(m, u, b)           # u ← b
+    kcopy!(m, u, b)              # u ← b
     kfill!(u_prev, zero(FC))
-    mul!(v, Aᴴ, u)            # v₁ ← Aᴴ * b
-    β = sqrt(kdotr(n, v, v))  # β₁ = v₁ᵀ M v₁
+    mul!(v, Aᴴ, u)               # v₁ ← Aᴴ * b
+    β = knorm_elliptic(n, v, v)  # β₁ = v₁ᵀ M v₁
     kfill!(rNorms, β)
     if history
       for i = 1 : nshifts
@@ -203,7 +203,7 @@ kwargs_cgls_lanczos_shift = (:M, :ldiv, :atol, :rtol, :itmax, :timemax, :verbose
       kaxpy!(m, -δ, u, utilde)        # uₖ₊₁ = utildeₖ - δₖuₖ - βₖuₖ₋₁
       kaxpy!(m, -β, u_prev, utilde)
       mul!(v, Aᴴ, utilde)             # vₖ₊₁ = Aᴴuₖ₊₁
-      β = sqrt(kdotr(n, v, v))        # βₖ₊₁ = vₖ₊₁ᵀ M vₖ₊₁
+      β = knorm_elliptic(n, v, v)     # βₖ₊₁ = vₖ₊₁ᵀ M vₖ₊₁
       kscal!(n, one(FC) / β, v)       # vₖ₊₁  ←  vₖ₊₁ / βₖ₊₁
       kscal!(m, one(FC) / β, utilde)  # uₖ₊₁ = uₖ₊₁ / βₖ₊₁
       kcopy!(m, u_prev, u)            # u_prev ← u

diff --git a/src/cr.jl b/src/cr.jl
@@ -153,8 +153,8 @@ kwargs_cr = (:M, :ldiv, :radius, :linesearch, :γ, :atol, :rtol, :itmax, :timema
     mul!(Ar, A, r)
     ρ = kdotr(n, r, Ar)
 
-    rNorm = sqrt(kdotr(n, r, p))    # ‖r‖
-    history && push!(rNorms, rNorm) # Values of ‖r‖
+    rNorm = knorm_elliptic(n, r, p)  # ‖r‖
+    history && push!(rNorms, rNorm)  # Values of ‖r‖
 
     if ρ == 0
       stats.niter = 0

diff --git a/src/craig.jl b/src/craig.jl
@@ -204,7 +204,7 @@ kwargs_craig = (:M, :N, :ldiv, :transfer_to_lsqr, :sqd, :λ, :btol, :conlim, :at
 
     kcopy!(m, Mu, b)  # Mu ← b
     MisI || mulorldiv!(u, M, Mu, ldiv)
-    β₁ = sqrt(kdotr(m, u, Mu))
+    β₁ = knorm_elliptic(m, u, Mu)
     rNorm  = β₁
     history && push!(rNorms, rNorm)
     if β₁ == 0
@@ -270,7 +270,7 @@ kwargs_craig = (:M, :N, :ldiv, :transfer_to_lsqr, :sqd, :λ, :btol, :conlim, :at
       mul!(Aᴴu, Aᴴ, u)
       kaxpby!(n, one(FC), Aᴴu, -β, Nv)
       NisI || mulorldiv!(v, N, Nv, ldiv)
-      α = sqrt(kdotr(n, v, Nv))
+      α = knorm_elliptic(n, v, Nv)
       if α == 0
         inconsistent = true
         continue
@@ -313,7 +313,7 @@ kwargs_craig = (:M, :N, :ldiv, :transfer_to_lsqr, :sqd, :λ, :btol, :conlim, :at
       mul!(Av, A, v)
       kaxpby!(m, one(FC), Av, -α, Mu)
       MisI || mulorldiv!(u, M, Mu, ldiv)
-      β = sqrt(kdotr(m, u, Mu))
+      β = knorm_elliptic(m, u, Mu)
       if β ≠ 0
         kscal!(m, one(FC) / β, u)
         MisI || kscal!(m, one(FC) / β, Mu)

diff --git a/src/craigmr.jl b/src/craigmr.jl
@@ -191,7 +191,7 @@ kwargs_craigmr = (:M, :N, :ldiv, :sqd, :λ, :atol, :rtol, :itmax, :timemax, :ver
     kfill!(y, zero(FC))
     kcopy!(m, Mu, b)  # Mu ← b
     MisI || mulorldiv!(u, M, Mu, ldiv)
-    β = sqrt(kdotr(m, u, Mu))
+    β = knorm_elliptic(m, u, Mu)
     if β == 0
       stats.niter = 0
       stats.solved, stats.inconsistent = true, false
@@ -210,7 +210,7 @@ kwargs_craigmr = (:M, :N, :ldiv, :sqd, :λ, :atol, :rtol, :itmax, :timemax, :ver
     mul!(Aᴴu, Aᴴ, u)
     kcopy!(n, Nv, Aᴴu)  # Nv ← Aᴴu
     NisI || mulorldiv!(v, N, Nv, ldiv)
-    α = sqrt(kdotr(n, v, Nv))
+    α = knorm_elliptic(n, v, Nv)
     Anorm² = α * α
 
     iter = 0
@@ -277,7 +277,7 @@ kwargs_craigmr = (:M, :N, :ldiv, :sqd, :λ, :atol, :rtol, :itmax, :timemax, :ver
       mul!(Av, A, v)
       kaxpby!(m, one(FC), Av, -α, Mu)
       MisI || mulorldiv!(u, M, Mu, ldiv)
-      β = sqrt(kdotr(m, u, Mu))
+      β = knorm_elliptic(m, u, Mu)
       if β ≠ 0
         kscal!(m, one(FC)/β, u)
         MisI || kscal!(m, one(FC)/β, Mu)
@@ -339,7 +339,7 @@ kwargs_craigmr = (:M, :N, :ldiv, :sqd, :λ, :atol, :rtol, :itmax, :timemax, :ver
       mul!(Aᴴu, Aᴴ, u)
       kaxpby!(n, one(FC), Aᴴu, -β, Nv)
       NisI || mulorldiv!(v, N, Nv, ldiv)
-      α = sqrt(kdotr(n, v, Nv))
+      α = knorm_elliptic(n, v, Nv)
       Anorm² = Anorm² + α * α  # = ‖Lₖ‖
       ArNorm = α * β * abs(ζ/ρ)
       history && push!(ArNorms, ArNorm)

diff --git a/src/krylov_utils.jl b/src/krylov_utils.jl
@@ -315,6 +315,8 @@ kdotr(n :: Integer, x :: AbstractVector{Complex{T}}, y :: AbstractVector{Complex
 knorm(n :: Integer, x :: Vector{T}) where T <: BLAS.BlasFloat = BLAS.nrm2(n, x, 1)
 knorm(n :: Integer, x :: AbstractVector{T}) where T <: FloatOrComplex = norm(x)
 
+knorm_elliptic(n :: Integer, x :: AbstractVector{T}, y :: AbstractVector{T}) where T <: FloatOrComplex = (x === y) ? knorm(n, x) : sqrt(kdotr(n, x, y))
+
 kscal!(n :: Integer, s :: T, x :: Vector{T}) where T <: BLAS.BlasFloat = BLAS.scal!(n, s, x, 1)
 kscal!(n :: Integer, s :: T, x :: AbstractVector{T}) where T <: FloatOrComplex = (x .*= s)
 kscal!(n :: Integer, s :: T, x :: AbstractVector{Complex{T}}) where T <: AbstractFloat = kscal!(n, Complex{T}(s), x)

diff --git a/src/lnlq.jl b/src/lnlq.jl
@@ -228,7 +228,7 @@ kwargs_lnlq = (:M, :N, :ldiv, :transfer_to_craig, :sqd, :λ, :σ, :utolx, :utoly
     # β₁Mu₁ = b.
     kcopy!(m, Mu, b)  # Mu ← b
     MisI || mulorldiv!(u, M, Mu, ldiv)  # u₁ = M⁻¹ * Mu₁
-    βₖ = sqrt(kdotr(m, u, Mu))          # β₁ = ‖u₁‖_M
+    βₖ = knorm_elliptic(m, u, Mu)       # β₁ = ‖u₁‖_M
     if βₖ ≠ 0
       kscal!(m, one(FC) / βₖ, u)
       MisI || kscal!(m, one(FC) / βₖ, Mu)
@@ -238,7 +238,7 @@ kwargs_lnlq = (:M, :N, :ldiv, :transfer_to_craig, :sqd, :λ, :σ, :utolx, :utoly
     mul!(Aᴴu, Aᴴ, u)
     kcopy!(n, Nv, Aᴴu)  # Nv ← Aᴴu
     NisI || mulorldiv!(v, N, Nv, ldiv)  # v₁ = N⁻¹ * Nv₁
-    αₖ = sqrt(kdotr(n, v, Nv))          # α₁ = ‖v₁‖_N
+    αₖ = knorm_elliptic(n, v, Nv)       # α₁ = ‖v₁‖_N
     if αₖ ≠ 0
       kscal!(n, one(FC) / αₖ, v)
       NisI || kscal!(n, one(FC) / αₖ, Nv)
@@ -346,7 +346,7 @@ kwargs_lnlq = (:M, :N, :ldiv, :transfer_to_craig, :sqd, :λ, :σ, :utolx, :utoly
       mul!(Av, A, v)
       kaxpby!(m, one(FC), Av, -αₖ, Mu)
       MisI || mulorldiv!(u, M, Mu, ldiv)  # uₖ₊₁ = M⁻¹ * Muₖ₊₁
-      βₖ₊₁ = sqrt(kdotr(m, u, Mu))        # βₖ₊₁ = ‖uₖ₊₁‖_M
+      βₖ₊₁ = knorm_elliptic(m, u, Mu)     # βₖ₊₁ = ‖uₖ₊₁‖_M
       if βₖ₊₁ ≠ 0
         kscal!(m, one(FC) / βₖ₊₁, u)
         MisI || kscal!(m, one(FC) / βₖ₊₁, Mu)
@@ -356,7 +356,7 @@ kwargs_lnlq = (:M, :N, :ldiv, :transfer_to_craig, :sqd, :λ, :σ, :utolx, :utoly
       mul!(Aᴴu, Aᴴ, u)
       kaxpby!(n, one(FC), Aᴴu, -βₖ₊₁, Nv)
       NisI || mulorldiv!(v, N, Nv, ldiv)  # vₖ₊₁ = N⁻¹ * Nvₖ₊₁
-      αₖ₊₁ = sqrt(kdotr(n, v, Nv))        # αₖ₊₁ = ‖vₖ₊₁‖_N
+      αₖ₊₁ = knorm_elliptic(n, v, Nv)     # αₖ₊₁ = ‖vₖ₊₁‖_N
       if αₖ₊₁ ≠ 0
         kscal!(n, one(FC) / αₖ₊₁, v)
         NisI || kscal!(n, one(FC) / αₖ₊₁, Nv)

diff --git a/src/lslq.jl b/src/lslq.jl
@@ -228,7 +228,7 @@ kwargs_lslq = (:M, :N, :ldiv, :transfer_to_lsqr, :sqd, :λ, :σ, :etol, :utol, :
     # β₁ M u₁ = b.
     kcopy!(m, Mu, b)  # Mu ← b
     MisI || mulorldiv!(u, M, Mu, ldiv)
-    β₁ = sqrt(kdotr(m, u, Mu))
+    β₁ = knorm_elliptic(m, u, Mu)
     if β₁ == 0
       stats.niter = 0
       stats.solved, stats.inconsistent = true, false
@@ -246,7 +246,7 @@ kwargs_lslq = (:M, :N, :ldiv, :transfer_to_lsqr, :sqd, :λ, :σ, :etol, :utol, :
     mul!(Aᴴu, Aᴴ, u)
     kcopy!(n, Nv, Aᴴu)  # Nv ← Aᴴu
     NisI || mulorldiv!(v, N, Nv, ldiv)
-    α = sqrt(kdotr(n, v, Nv))  # = α₁
+    α = knorm_elliptic(n, v, Nv)  # = α₁
 
     # Aᴴb = 0 so x = 0 is a minimum least-squares solution
     if α == 0
@@ -326,7 +326,7 @@ kwargs_lslq = (:M, :N, :ldiv, :transfer_to_lsqr, :sqd, :λ, :σ, :etol, :utol, :
       mul!(Av, A, v)
       kaxpby!(m, one(FC), Av, -α, Mu)
       MisI || mulorldiv!(u, M, Mu, ldiv)
-      β = sqrt(kdotr(m, u, Mu))
+      β = knorm_elliptic(m, u, Mu)
       if β ≠ 0
         kscal!(m, one(FC)/β, u)
         MisI || kscal!(m, one(FC)/β, Mu)
@@ -335,7 +335,7 @@ kwargs_lslq = (:M, :N, :ldiv, :transfer_to_lsqr, :sqd, :λ, :σ, :etol, :utol, :
         mul!(Aᴴu, Aᴴ, u)
         kaxpby!(n, one(FC), Aᴴu, -β, Nv)
         NisI || mulorldiv!(v, N, Nv, ldiv)
-        α = sqrt(kdotr(n, v, Nv))
+        α = knorm_elliptic(n, v, Nv)
         if α ≠ 0
           kscal!(n, one(FC)/α, v)
           NisI || kscal!(n, one(FC)/α, Nv)

diff --git a/src/lsmr.jl b/src/lsmr.jl
@@ -202,7 +202,7 @@ kwargs_lsmr = (:M, :N, :ldiv, :sqd, :λ, :radius, :etol, :axtol, :btol, :conlim,
     # β₁ M u₁ = b.
     kcopy!(m, Mu, b)  # Mu ← b
     MisI || mulorldiv!(u, M, Mu, ldiv)
-    β₁ = sqrt(kdotr(m, u, Mu))
+    β₁ = knorm_elliptic(m, u, Mu)
     if β₁ == 0
       stats.niter = 0
       stats.solved, stats.inconsistent = true, false
@@ -219,7 +219,7 @@ kwargs_lsmr = (:M, :N, :ldiv, :sqd, :λ, :radius, :etol, :axtol, :btol, :conlim,
     mul!(Aᴴu, Aᴴ, u)
     kcopy!(n, Nv, Aᴴu)  # Nv ← Aᴴu
     NisI || mulorldiv!(v, N, Nv, ldiv)
-    α = sqrt(kdotr(n, v, Nv))
+    α = knorm_elliptic(n, v, Nv)
 
     ζbar = α * β
     αbar = α
@@ -295,7 +295,7 @@ kwargs_lsmr = (:M, :N, :ldiv, :sqd, :λ, :radius, :etol, :axtol, :btol, :conlim,
       mul!(Av, A, v)
       kaxpby!(m, one(FC), Av, -α, Mu)
       MisI || mulorldiv!(u, M, Mu, ldiv)
-      β = sqrt(kdotr(m, u, Mu))
+      β = knorm_elliptic(m, u, Mu)
       if β ≠ 0
         kscal!(m, one(FC)/β, u)
         MisI || kscal!(m, one(FC)/β, Mu)
@@ -304,7 +304,7 @@ kwargs_lsmr = (:M, :N, :ldiv, :sqd, :λ, :radius, :etol, :axtol, :btol, :conlim,
         mul!(Aᴴu, Aᴴ, u)
         kaxpby!(n, one(FC), Aᴴu, -β, Nv)
         NisI || mulorldiv!(v, N, Nv, ldiv)
-        α = sqrt(kdotr(n, v, Nv))
+        α = knorm_elliptic(n, v, Nv)
         if α ≠ 0
           kscal!(n, one(FC)/α, v)
           NisI || kscal!(n, one(FC)/α, Nv)

diff --git a/src/lsqr.jl b/src/lsqr.jl
@@ -199,7 +199,7 @@ kwargs_lsqr = (:M, :N, :ldiv, :sqd, :λ, :radius, :etol, :axtol, :btol, :conlim,
     # β₁ M u₁ = b.
     kcopy!(m, Mu, b)  # Mu ← b
     MisI || mulorldiv!(u, M, Mu, ldiv)
-    β₁ = sqrt(kdotr(m, u, Mu))
+    β₁ = knorm_elliptic(m, u, Mu)
     if β₁ == 0
       stats.niter = 0
       stats.solved, stats.inconsistent = true, false
@@ -283,7 +283,7 @@ kwargs_lsqr = (:M, :N, :ldiv, :sqd, :λ, :radius, :etol, :axtol, :btol, :conlim,
       mul!(Av, A, v)
       kaxpby!(m, one(FC), Av, -α, Mu)
       MisI || mulorldiv!(u, M, Mu, ldiv)
-      β = sqrt(kdotr(m, u, Mu))
+      β = knorm_elliptic(m, u, Mu)
       if β ≠ 0
         kscal!(m, one(FC)/β, u)
         MisI || kscal!(m, one(FC)/β, Mu)
@@ -294,7 +294,7 @@ kwargs_lsqr = (:M, :N, :ldiv, :sqd, :λ, :radius, :etol, :axtol, :btol, :conlim,
         mul!(Aᴴu, Aᴴ, u)
         kaxpby!(n, one(FC), Aᴴu, -β, Nv)
         NisI || mulorldiv!(v, N, Nv, ldiv)
-        α = sqrt(kdotr(n, v, Nv))
+        α = knorm_elliptic(n, v, Nv)
         if α ≠ 0
           kscal!(n, one(FC)/α, v)
           NisI || kscal!(n, one(FC)/α, Nv)

diff --git a/src/minres_qlp.jl b/src/minres_qlp.jl
@@ -151,7 +151,7 @@ kwargs_minres_qlp = (:M, :ldiv, :λ, :atol, :rtol, :Artol, :itmax, :timemax, :ve
 
     # β₁v₁ = Mb
     MisI || mulorldiv!(vₖ, M, M⁻¹vₖ, ldiv)
-    βₖ = sqrt(kdotr(n, vₖ, M⁻¹vₖ))
+    βₖ = knorm_elliptic(n, vₖ, M⁻¹vₖ)
     if βₖ ≠ 0
       kscal!(n, one(FC) / βₖ, M⁻¹vₖ)
       MisI || kscal!(n, one(FC) / βₖ, vₖ)
@@ -231,7 +231,7 @@ kwargs_minres_qlp = (:M, :ldiv, :λ, :atol, :rtol, :Artol, :itmax, :timemax, :ve
 
       MisI || mulorldiv!(vₖ₊₁, M, p, ldiv)  # βₖ₊₁vₖ₊₁ = MAvₖ - γₖvₖ₋₁ - αₖvₖ
 
-      βₖ₊₁ = sqrt(kdotr(m, vₖ₊₁, p))
+      βₖ₊₁ = knorm_elliptic(m, vₖ₊₁, p)
 
       # βₖ₊₁.ₖ ≠ 0
       if βₖ₊₁ > btol

diff --git a/src/tricg.jl b/src/tricg.jl
@@ -214,7 +214,7 @@ kwargs_tricg = (:M, :N, :ldiv, :spd, :snd, :flip, :τ, :ν, :atol, :rtol, :itmax
     # β₁Ev₁ = b ↔ β₁v₁ = Mb
     kcopy!(m, M⁻¹vₖ, b₀)  # M⁻¹vₖ ← b₀
     MisI || mulorldiv!(vₖ, M, M⁻¹vₖ, ldiv)
-    βₖ = sqrt(kdotr(m, vₖ, M⁻¹vₖ))  # β₁ = ‖v₁‖_E
+    βₖ = knorm_elliptic(m, vₖ, M⁻¹vₖ)  # β₁ = ‖v₁‖_E
     if βₖ ≠ 0
       kscal!(m, one(FC) / βₖ, M⁻¹vₖ)
       MisI || kscal!(m, one(FC) / βₖ, vₖ)
@@ -225,7 +225,7 @@ kwargs_tricg = (:M, :N, :ldiv, :spd, :snd, :flip, :τ, :ν, :atol, :rtol, :itmax
     # γ₁Fu₁ = c ↔ γ₁u₁ = Nc
     kcopy!(n, N⁻¹uₖ, c₀)  # M⁻¹uₖ ← c₀
     NisI || mulorldiv!(uₖ, N, N⁻¹uₖ, ldiv)
-    γₖ = sqrt(kdotr(n, uₖ, N⁻¹uₖ))  # γ₁ = ‖u₁‖_F
+    γₖ = knorm_elliptic(n, uₖ, N⁻¹uₖ))  # γ₁ = ‖u₁‖_F
     if γₖ ≠ 0
       kscal!(n, one(FC) / γₖ, N⁻¹uₖ)
       NisI || kscal!(n, one(FC) / γₖ, uₖ)
@@ -382,8 +382,8 @@ kwargs_tricg = (:M, :N, :ldiv, :spd, :snd, :flip, :τ, :ν, :atol, :rtol, :itmax
       MisI || mulorldiv!(vₖ₊₁, M, q, ldiv)  # βₖ₊₁vₖ₊₁ = MAuₖ  - γₖvₖ₋₁ - αₖvₖ
       NisI || mulorldiv!(uₖ₊₁, N, p, ldiv)  # γₖ₊₁uₖ₊₁ = NAᴴvₖ - βₖuₖ₋₁ - ᾱₖuₖ
 
-      βₖ₊₁ = sqrt(kdotr(m, vₖ₊₁, q))  # βₖ₊₁ = ‖vₖ₊₁‖_E
-      γₖ₊₁ = sqrt(kdotr(n, uₖ₊₁, p))  # γₖ₊₁ = ‖uₖ₊₁‖_F
+      βₖ₊₁ = knorm_elliptic(m, vₖ₊₁, q)  # βₖ₊₁ = ‖vₖ₊₁‖_E
+      γₖ₊₁ = knorm_elliptic(n, uₖ₊₁, p)  # γₖ₊₁ = ‖uₖ₊₁‖_F
 
       # βₖ₊₁ ≠ 0
       if βₖ₊₁ > btol

diff --git a/src/trimr.jl b/src/trimr.jl
@@ -220,7 +220,7 @@ kwargs_trimr = (:M, :N, :ldiv, :spd, :snd, :flip, :sp, :τ, :ν, :atol, :rtol, :
     # β₁Ev₁ = b ↔ β₁v₁ = Mb
     kcopy!(m, M⁻¹vₖ, b₀)  # M⁻¹vₖ ← b₀
     MisI || mulorldiv!(vₖ, M, M⁻¹vₖ, ldiv)
-    βₖ = sqrt(kdotr(m, vₖ, M⁻¹vₖ))  # β₁ = ‖v₁‖_E
+    βₖ = knorm_elliptic(m, vₖ, M⁻¹vₖ)  # β₁ = ‖v₁‖_E
     if βₖ ≠ 0
       kscal!(m, one(FC) / βₖ, M⁻¹vₖ)
       MisI || kscal!(m, one(FC) / βₖ, vₖ)
@@ -231,7 +231,7 @@ kwargs_trimr = (:M, :N, :ldiv, :spd, :snd, :flip, :sp, :τ, :ν, :atol, :rtol, :
     # γ₁Fu₁ = c ↔ γ₁u₁ = Nc
     kcopy!(n, N⁻¹uₖ, c₀)  # N⁻¹uₖ ← c₀
     NisI || mulorldiv!(uₖ, N, N⁻¹uₖ, ldiv)
-    γₖ = sqrt(kdotr(n, uₖ, N⁻¹uₖ))  # γ₁ = ‖u₁‖_F
+    γₖ = knorm_elliptic(n, uₖ, N⁻¹uₖ)  # γ₁ = ‖u₁‖_F
     if γₖ ≠ 0
       kscal!(n, one(FC) / γₖ, N⁻¹uₖ)
       NisI || kscal!(n, one(FC) / γₖ, uₖ)
@@ -302,8 +302,8 @@ kwargs_trimr = (:M, :N, :ldiv, :spd, :snd, :flip, :sp, :τ, :ν, :atol, :rtol, :
       MisI || mulorldiv!(vₖ₊₁, M, q, ldiv)  # βₖ₊₁vₖ₊₁ = MAuₖ  - γₖvₖ₋₁ - αₖvₖ
       NisI || mulorldiv!(uₖ₊₁, N, p, ldiv)  # γₖ₊₁uₖ₊₁ = NAᴴvₖ - βₖuₖ₋₁ - ᾱₖuₖ
 
-      βₖ₊₁ = sqrt(kdotr(m, vₖ₊₁, q))  # βₖ₊₁ = ‖vₖ₊₁‖_E
-      γₖ₊₁ = sqrt(kdotr(n, uₖ₊₁, p))  # γₖ₊₁ = ‖uₖ₊₁‖_F
+      βₖ₊₁ = knorm_elliptic(m, vₖ₊₁, q)  # βₖ₊₁ = ‖vₖ₊₁‖_E
+      γₖ₊₁ = knorm_elliptic(n, uₖ₊₁, p)  # γₖ₊₁ = ‖uₖ₊₁‖_F
 
       # βₖ₊₁ ≠ 0
       if βₖ₊₁ > btol