[code] Use Aᴴ instead of Aᵀ

src/bicgstab.jl

@@ -26,10 +26,10 @@ export bicgstab, bicgstab!
 
 Solve the square linear system Ax = b using the BICGSTAB method.
 BICGSTAB requires two initial vectors `b` and `c`.
-The relation `bᵀc ≠ 0` must be satisfied and by default `c = b`.
+The relation `bᴴc ≠ 0` must be satisfied and by default `c = b`.
 
 The Biconjugate Gradient Stabilized method is a variant of BiCG, like CGS,
-but using different updates for the Aᵀ-sequence in order to obtain smoother
+but using different updates for the Aᴴ-sequence in order to obtain smoother
 convergence than CGS.
 
 If BICGSTAB stagnates, we recommend DQGMRES and BiLQ as alternative methods for unsymmetric square systems.
@@ -157,7 +157,7 @@ function bicgstab!(solver :: BicgstabSolver{T,FC,S}, A, b :: AbstractVector{FC};
   if next_ρ == 0
     stats.niter = 0
     stats.solved, stats.inconsistent = false, false
-    stats.status = "Breakdown bᵀc = 0"
+    stats.status = "Breakdown bᴴc = 0"
     solver.warm_start = false
     return solver
   end

src/bilq.jl

@@ -24,7 +24,7 @@ export bilq, bilq!
 Solve the square linear system Ax = b using the BiLQ method.
 
 BiLQ is based on the Lanczos biorthogonalization process and requires two initial vectors `b` and `c`.
-The relation `bᵀc ≠ 0` must be satisfied and by default `c = b`.
+The relation `bᴴc ≠ 0` must be satisfied and by default `c = b`.
 When `A` is symmetric and `b = c`, BiLQ is equivalent to SYMMLQ.
 
 An option gives the possibility of transferring to the BiCG point,
@@ -90,7 +90,7 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
   ktypeof(c) == S || error("ktypeof(c) ≠ $S")
 
   # Compute the adjoint of A
-  Aᵀ = A'
+  Aᴴ = A'
 
   # Set up workspace.
   uₖ₋₁, uₖ, q, vₖ₋₁, vₖ = solver.uₖ₋₁, solver.uₖ, solver.q, solver.vₖ₋₁, solver.vₖ
@@ -127,25 +127,25 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
   kdisplay(iter, verbose) && @printf("%5d  %7.1e\n", iter, bNorm)
 
   # Initialize the Lanczos biorthogonalization process.
-  cᵗb = @kdot(n, c, r₀)  # ⟨c,r₀⟩
-  if cᵗb == 0
+  cᴴb = @kdot(n, c, r₀)  # ⟨c,r₀⟩
+  if cᴴb == 0
     stats.niter = 0
     stats.solved = false
     stats.inconsistent = false
-    stats.status = "Breakdown bᵀc = 0"
+    stats.status = "Breakdown bᴴc = 0"
     solver.warm_start = false
     return solver
   end
 
-  βₖ = √(abs(cᵗb))            # β₁γ₁ = cᵀ(b - Ax₀)
-  γₖ = cᵗb / βₖ               # β₁γ₁ = cᵀ(b - Ax₀)
+  βₖ = √(abs(cᴴb))            # β₁γ₁ = cᴴ(b - Ax₀)
+  γₖ = cᴴb / βₖ               # β₁γ₁ = cᴴ(b - Ax₀)
   vₖ₋₁ .= zero(FC)            # v₀ = 0
   uₖ₋₁ .= zero(FC)            # u₀ = 0
   vₖ .= r₀ ./ βₖ              # v₁ = (b - Ax₀) / β₁
   uₖ .= c ./ conj(γₖ)         # u₁ = c / γ̄₁
   cₖ₋₁ = cₖ = -one(T)         # Givens cosines used for the LQ factorization of Tₖ
   sₖ₋₁ = sₖ = zero(FC)        # Givens sines used for the LQ factorization of Tₖ
-  d̅ .= zero(FC)               # Last column of D̅ₖ = Vₖ(Qₖ)ᵀ
+  d̅ .= zero(FC)               # Last column of D̅ₖ = Vₖ(Qₖ)ᴴ
   ζₖ₋₁ = ζbarₖ = zero(FC)     # ζₖ₋₁ and ζbarₖ are the last components of z̅ₖ = (L̅ₖ)⁻¹β₁e₁
   ζₖ₋₂ = ηₖ = zero(FC)        # ζₖ₋₂ and ηₖ are used to update ζₖ₋₁ and ζbarₖ
   δbarₖ₋₁ = δbarₖ = zero(FC)  # Coefficients of Lₖ₋₁ and L̅ₖ modified over the course of two iterations
@@ -165,10 +165,10 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
 
     # Continue the Lanczos biorthogonalization process.
     # AVₖ  = VₖTₖ    + βₖ₊₁vₖ₊₁(eₖ)ᵀ = Vₖ₊₁Tₖ₊₁.ₖ
-    # AᵀUₖ = Uₖ(Tₖ)ᵀ + γ̄ₖ₊₁uₖ₊₁(eₖ)ᵀ = Uₖ₊₁(Tₖ.ₖ₊₁)ᵀ
+    # AᴴUₖ = Uₖ(Tₖ)ᴴ + γ̄ₖ₊₁uₖ₊₁(eₖ)ᵀ = Uₖ₊₁(Tₖ.ₖ₊₁)ᴴ
 
     mul!(q, A , vₖ)  # Forms vₖ₊₁ : q ← Avₖ
-    mul!(p, Aᵀ, uₖ)  # Forms uₖ₊₁ : p ← Aᵀuₖ
+    mul!(p, Aᴴ, uₖ)  # Forms uₖ₊₁ : p ← Aᴴuₖ
 
     @kaxpy!(n, -γₖ, vₖ₋₁, q)  # q ← q - γₖ * vₖ₋₁
     @kaxpy!(n, -βₖ, uₖ₋₁, p)  # p ← p - β̄ₖ * uₖ₋₁
@@ -178,9 +178,9 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
     @kaxpy!(n, -     αₖ , vₖ, q)    # q ← q - αₖ * vₖ
     @kaxpy!(n, -conj(αₖ), uₖ, p)    # p ← p - ᾱₖ * uₖ
 
-    pᵗq = @kdot(n, p, q)      # pᵗq  = ⟨p,q⟩
-    βₖ₊₁ = √(abs(pᵗq))        # βₖ₊₁ = √(|pᵗq|)
-    γₖ₊₁ = pᵗq / βₖ₊₁         # γₖ₊₁ = pᵗq / βₖ₊₁
+    pᴴq = @kdot(n, p, q)      # pᴴq  = ⟨p,q⟩
+    βₖ₊₁ = √(abs(pᴴq))        # βₖ₊₁ = √(|pᴴq|)
+    γₖ₊₁ = pᴴq / βₖ₊₁         # γₖ₊₁ = pᴴq / βₖ₊₁
 
     # Update the LQ factorization of Tₖ = L̅ₖQₖ.
     # [ α₁ γ₂ 0  •  •  •  0 ]   [ δ₁   0    •   •   •    •    0   ]
@@ -235,7 +235,7 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
       ηₖ   = -ϵₖ₋₂ * ζₖ₋₂ - λₖ₋₁ * ζₖ₋₁
     end
 
-    # Relations for the directions dₖ₋₁ and d̅ₖ, the last two columns of D̅ₖ = Vₖ(Qₖ)ᵀ.
+    # Relations for the directions dₖ₋₁ and d̅ₖ, the last two columns of D̅ₖ = Vₖ(Qₖ)ᴴ.
     # [d̅ₖ₋₁ vₖ] [cₖ  s̄ₖ] = [dₖ₋₁ d̅ₖ] ⟷ dₖ₋₁ = cₖ * d̅ₖ₋₁ + sₖ * vₖ
     #           [sₖ -cₖ]             ⟷ d̅ₖ   = s̄ₖ * d̅ₖ₋₁ - cₖ * vₖ
     if iter ≥ 2
@@ -258,13 +258,13 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
     @. vₖ₋₁ = vₖ # vₖ₋₁ ← vₖ
     @. uₖ₋₁ = uₖ # uₖ₋₁ ← uₖ
 
-    if pᵗq ≠ 0
+    if pᴴq ≠ 0
       @. 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‖
-    vₖᵀvₖ₊₁ = @kdot(n, vₖ₋₁, vₖ)
+    vₖᴴvₖ₊₁ = @kdot(n, vₖ₋₁, vₖ)
     norm_vₖ₊₁ = @knrm2(n, vₖ)
 
     # Compute BiLQ residual norm
@@ -274,7 +274,7 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
     else
       μₖ = βₖ * (sₖ₋₁ * ζₖ₋₂ - cₖ₋₁ * cₖ * ζₖ₋₁) + αₖ * sₖ * ζₖ₋₁
       ωₖ = βₖ₊₁ * sₖ * ζₖ₋₁
-      θₖ = conj(μₖ) * ωₖ * vₖᵀvₖ₊₁
+      θₖ = conj(μₖ) * ωₖ * vₖᴴvₖ₊₁
       rNorm_lq = sqrt(abs2(μₖ) * norm_vₖ^2 + abs2(ωₖ) * norm_vₖ₊₁^2 + 2 * real(θₖ))
     end
     history && push!(rNorms, rNorm_lq)
@@ -300,7 +300,7 @@ function bilq!(solver :: BilqSolver{T,FC,S}, A, b :: AbstractVector{FC}; c :: Ab
     solved_lq = rNorm_lq ≤ ε
     solved_cg = transfer_to_bicg && (abs(δbarₖ) > eps(T)) && (rNorm_cg ≤ ε)
     tired = iter ≥ itmax
-    breakdown = !solved_lq && !solved_cg && (pᵗq == 0)
+    breakdown = !solved_lq && !solved_cg && (pᴴq == 0)
     kdisplay(iter, verbose) && @printf("%5d  %7.1e\n", iter, rNorm_lq)
   end
   (verbose > 0) && @printf("\n")

src/bilqr.jl

@@ -1,5 +1,5 @@
 # An implementation of BILQR for the solution of square
-# consistent linear adjoint systems Ax = b and Aᵀy = c.
+# consistent linear adjoint systems Ax = b and Aᴴy = c.
 #
 # This method is described in
 #
@@ -24,11 +24,11 @@ export bilqr, bilqr!
 Combine BiLQ and QMR to solve adjoint systems.
 
     [0  A] [y] = [b]
-    [Aᵀ 0] [x]   [c]
+    [Aᴴ 0] [x]   [c]
 
-The relation `bᵀc ≠ 0` must be satisfied.
+The relation `bᴴc ≠ 0` must be satisfied.
 BiLQ is used for solving primal system `Ax = b`.
-QMR is used for solving dual system `Aᵀy = c`.
+QMR is used for solving dual system `Aᴴy = c`.
 
 An option gives the possibility of transferring from the BiLQ point to the
 BiCG point, when it exists. The transfer is based on the residual norm.
@@ -94,7 +94,7 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
   ktypeof(c) == S || error("ktypeof(c) ≠ $S")
 
   # Compute the adjoint of A
-  Aᵀ = A'
+  Aᴴ = A'
 
   # Set up workspace.
   uₖ₋₁, uₖ, q, vₖ₋₁, vₖ = solver.uₖ₋₁, solver.uₖ, solver.q, solver.vₖ₋₁, solver.vₖ
@@ -109,15 +109,15 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
   if warm_start
     mul!(r₀, A, Δx)
     @kaxpby!(n, one(FC), b, -one(FC), r₀)
-    mul!(s₀, Aᵀ, Δy)
+    mul!(s₀, Aᴴ, Δy)
     @kaxpby!(n, one(FC), c, -one(FC), s₀)
   end
 
   # Initial solution x₀ and residual norm ‖r₀‖ = ‖b - Ax₀‖.
   x .= zero(FC)          # x₀
   bNorm = @knrm2(n, r₀)  # rNorm = ‖r₀‖
 
-  # Initial solution t₀ and residual norm ‖s₀‖ = ‖c - Aᵀy₀‖.
+  # Initial solution t₀ and residual norm ‖s₀‖ = ‖c - Aᴴy₀‖.
   t .= zero(FC)          # t₀
   cNorm = @knrm2(n, s₀)  # sNorm = ‖s₀‖
 
@@ -132,34 +132,34 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
   kdisplay(iter, verbose) && @printf("%5d  %7.1e  %7.1e\n", iter, bNorm, cNorm)
 
   # Initialize the Lanczos biorthogonalization process.
-  cᵗb = @kdot(n, s₀, r₀)  # ⟨s₀,r₀⟩ = ⟨c - Aᵀy₀,b - Ax₀⟩
-  if cᵗb == 0
+  cᴴb = @kdot(n, s₀, r₀)  # ⟨s₀,r₀⟩ = ⟨c - Aᴴy₀,b - Ax₀⟩
+  if cᴴb == 0
     stats.niter = 0
     stats.solved_primal = false
     stats.solved_dual = false
-    stats.status = "Breakdown bᵀc = 0"
+    stats.status = "Breakdown bᴴc = 0"
     solver.warm_start = false
     return solver
   end
 
   # Set up workspace.
-  βₖ = √(abs(cᵗb))            # β₁γ₁ = (c - Aᵀy₀)ᵀ(b - Ax₀)
-  γₖ = cᵗb / βₖ               # β₁γ₁ = (c - Aᵀy₀)ᵀ(b - Ax₀)
+  βₖ = √(abs(cᴴb))            # β₁γ₁ = (c - Aᴴy₀)ᴴ(b - Ax₀)
+  γₖ = cᴴb / βₖ               # β₁γ₁ = (c - Aᴴy₀)ᴴ(b - Ax₀)
   vₖ₋₁ .= zero(FC)            # v₀ = 0
   uₖ₋₁ .= zero(FC)            # u₀ = 0
   vₖ .= r₀ ./ βₖ              # v₁ = (b - Ax₀) / β₁
-  uₖ .= s₀ ./ conj(γₖ)        # u₁ = (c - Aᵀy₀) / γ̄₁
+  uₖ .= s₀ ./ conj(γₖ)        # u₁ = (c - Aᴴy₀) / γ̄₁
   cₖ₋₁ = cₖ = -one(T)         # Givens cosines used for the LQ factorization of Tₖ
   sₖ₋₁ = sₖ = zero(FC)        # Givens sines used for the LQ factorization of Tₖ
-  d̅ .= zero(FC)               # Last column of D̅ₖ = Vₖ(Qₖ)ᵀ
+  d̅ .= zero(FC)               # Last column of D̅ₖ = Vₖ(Qₖ)ᴴ
   ζₖ₋₁ = ζbarₖ = zero(FC)     # ζₖ₋₁ and ζbarₖ are the last components of z̅ₖ = (L̅ₖ)⁻¹β₁e₁
   ζₖ₋₂ = ηₖ = zero(FC)        # ζₖ₋₂ and ηₖ are used to update ζₖ₋₁ and ζbarₖ
   δbarₖ₋₁ = δbarₖ = zero(FC)  # Coefficients of Lₖ₋₁ and L̅ₖ modified over the course of two iterations
   ψbarₖ₋₁ = ψₖ₋₁ = zero(FC)   # ψₖ₋₁ and ψbarₖ are the last components of h̅ₖ = Qₖγ̄₁e₁
   norm_vₖ = bNorm / βₖ        # ‖vₖ‖ is used for residual norm estimates
   ϵₖ₋₃ = λₖ₋₂ = zero(FC)      # Components of Lₖ₋₁
-  wₖ₋₃ .= zero(FC)            # Column k-3 of Wₖ = Uₖ(Lₖ)⁻ᵀ
-  wₖ₋₂ .= zero(FC)            # Column k-2 of Wₖ = Uₖ(Lₖ)⁻ᵀ
+  wₖ₋₃ .= zero(FC)            # Column k-3 of Wₖ = Uₖ(Lₖ)⁻ᴴ
+  wₖ₋₂ .= zero(FC)            # Column k-2 of Wₖ = Uₖ(Lₖ)⁻ᴴ
   τₖ = zero(T)                # τₖ is used for the dual residual norm estimate
 
   # Stopping criterion.
@@ -180,10 +180,10 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
 
     # Continue the Lanczos biorthogonalization process.
     # AVₖ  = VₖTₖ    + βₖ₊₁vₖ₊₁(eₖ)ᵀ = Vₖ₊₁Tₖ₊₁.ₖ
-    # AᵀUₖ = Uₖ(Tₖ)ᵀ + γ̄ₖ₊₁uₖ₊₁(eₖ)ᵀ = Uₖ₊₁(Tₖ.ₖ₊₁)ᵀ
+    # AᴴUₖ = Uₖ(Tₖ)ᴴ + γ̄ₖ₊₁uₖ₊₁(eₖ)ᵀ = Uₖ₊₁(Tₖ.ₖ₊₁)ᴴ
 
     mul!(q, A , vₖ)  # Forms vₖ₊₁ : q ← Avₖ
-    mul!(p, Aᵀ, uₖ)  # Forms uₖ₊₁ : p ← Aᵀuₖ
+    mul!(p, Aᴴ, uₖ)  # Forms uₖ₊₁ : p ← Aᴴuₖ
 
     @kaxpy!(n, -γₖ, vₖ₋₁, q)  # q ← q - γₖ * vₖ₋₁
     @kaxpy!(n, -βₖ, uₖ₋₁, p)  # p ← p - β̄ₖ * uₖ₋₁
@@ -193,9 +193,9 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
     @kaxpy!(n, -     αₖ , vₖ, q)  # q ← q - αₖ * vₖ
     @kaxpy!(n, -conj(αₖ), uₖ, p)  # p ← p - ᾱₖ * uₖ
 
-    pᵗq = @kdot(n, p, q)  # pᵗq  = ⟨p,q⟩
-    βₖ₊₁ = √(abs(pᵗq))    # βₖ₊₁ = √(|pᵗq|)
-    γₖ₊₁ = pᵗq / βₖ₊₁     # γₖ₊₁ = pᵗq / βₖ₊₁
+    pᴴq = @kdot(n, p, q)  # pᴴq  = ⟨p,q⟩
+    βₖ₊₁ = √(abs(pᴴq))    # βₖ₊₁ = √(|pᴴq|)
+    γₖ₊₁ = pᴴq / βₖ₊₁     # γₖ₊₁ = pᴴq / βₖ₊₁
 
     # Update the LQ factorization of Tₖ = L̅ₖQₖ.
     # [ α₁ γ₂ 0  •  •  •  0 ]   [ δ₁   0    •   •   •    •    0   ]
@@ -251,7 +251,7 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
         ηₖ   = -ϵₖ₋₂ * ζₖ₋₂ - λₖ₋₁ * ζₖ₋₁
       end
 
-      # Relations for the directions dₖ₋₁ and d̅ₖ, the last two columns of D̅ₖ = Vₖ(Qₖ)ᵀ.
+      # Relations for the directions dₖ₋₁ and d̅ₖ, the last two columns of D̅ₖ = Vₖ(Qₖ)ᴴ.
       # [d̅ₖ₋₁ vₖ] [cₖ  s̄ₖ] = [dₖ₋₁ d̅ₖ] ⟷ dₖ₋₁ = cₖ * d̅ₖ₋₁ + sₖ * vₖ
       #           [sₖ -cₖ]             ⟷ d̅ₖ   = s̄ₖ * d̅ₖ₋₁ - cₖ * vₖ
       if iter ≥ 2
@@ -271,7 +271,7 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
       end
 
       # Compute ⟨vₖ,vₖ₊₁⟩ and ‖vₖ₊₁‖
-      vₖᵀvₖ₊₁ = @kdot(n, vₖ, q) / βₖ₊₁
+      vₖᴴvₖ₊₁ = @kdot(n, vₖ, q) / βₖ₊₁
       norm_vₖ₊₁ = @knrm2(n, q) / βₖ₊₁
 
       # Compute BiLQ residual norm
@@ -281,7 +281,7 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
       else
         μₖ = βₖ * (sₖ₋₁ * ζₖ₋₂ - cₖ₋₁ * cₖ * ζₖ₋₁) + αₖ * sₖ * ζₖ₋₁
         ωₖ = βₖ₊₁ * sₖ * ζₖ₋₁
-        θₖ = conj(μₖ) * ωₖ * vₖᵀvₖ₊₁
+        θₖ = conj(μₖ) * ωₖ * vₖᴴvₖ₊₁
         rNorm_lq = sqrt(abs2(μₖ) * norm_vₖ^2 + abs2(ωₖ) * norm_vₖ₊₁^2 + 2 * real(θₖ))
       end
       history && push!(rNorms, rNorm_lq)
@@ -318,7 +318,7 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
         ψbarₖ = sₖ * ψbarₖ₋₁
       end
 
-      # Compute the direction wₖ₋₁, the last column of Wₖ₋₁ = (Uₖ₋₁)(Lₖ₋₁)⁻ᵀ ⟷ (L̄ₖ₋₁)(Wₖ₋₁)ᵀ = (Uₖ₋₁)ᵀ.
+      # Compute the direction wₖ₋₁, the last column of Wₖ₋₁ = (Uₖ₋₁)(Lₖ₋₁)⁻ᴴ ⟷ (L̄ₖ₋₁)(Wₖ₋₁)ᵀ = (Uₖ₋₁)ᵀ.
       # w₁ = u₁ / δ̄₁
       if iter == 2
         wₖ₋₁ = wₖ₋₂
@@ -372,7 +372,7 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
     @. vₖ₋₁ = vₖ  # vₖ₋₁ ← vₖ
     @. uₖ₋₁ = uₖ  # uₖ₋₁ ← uₖ
 
-    if pᵗq ≠ zero(FC)
+    if pᴴq ≠ zero(FC)
       @. vₖ = q / βₖ₊₁        # βₖ₊₁vₖ₊₁ = q
       @. uₖ = p / conj(γₖ₊₁)  # γ̄ₖ₊₁uₖ₊₁ = p
     end
@@ -392,7 +392,7 @@ function bilqr!(solver :: BilqrSolver{T,FC,S}, A, b :: AbstractVector{FC}, c ::
 
     user_requested_exit = callback(solver) :: Bool
     tired = iter ≥ itmax
-    breakdown = !solved_lq && !solved_cg && (pᵗq == 0)
+    breakdown = !solved_lq && !solved_cg && (pᴴq == 0)
 
     kdisplay(iter, verbose) &&  solved_primal && !solved_dual && @printf("%5d  %7s  %7.1e\n", iter, "", sNorm)
     kdisplay(iter, verbose) && !solved_primal &&  solved_dual && @printf("%5d  %7.1e  %7s\n", iter, rNorm_lq, "")

src/cg_lanczos.jl

@@ -111,7 +111,7 @@ function cg_lanczos!(solver :: CgLanczosSolver{T,FC,S}, A, b :: AbstractVector{F
     Mv .= b
   end
   MisI || mulorldiv!(v, M, Mv, ldiv)  # v₁ = M⁻¹r₀
-  β = sqrt(@kdotr(n, v, Mv))          # β₁ = v₁ᵀ M v₁
+  β = sqrt(@kdotr(n, v, Mv))          # β₁ = v₁ᴴ M v₁
   σ = β
   rNorm = σ
   history && push!(rNorms, rNorm)
@@ -157,10 +157,10 @@ function cg_lanczos!(solver :: CgLanczosSolver{T,FC,S}, A, b :: AbstractVector{F
     # Form next Lanczos vector.
     # βₖ₊₁Mvₖ₊₁ = Avₖ - δₖMvₖ - βₖMvₖ₋₁
     mul!(Mv_next, A, v)        # Mvₖ₊₁ ← Avₖ
-    δ = @kdotr(n, v, Mv_next)  # δₖ = vₖᵀ A vₖ
+    δ = @kdotr(n, v, Mv_next)  # δₖ = vₖᴴ A vₖ
 
     # Check curvature. Exit fast if requested.
-    # It is possible to show that σₖ² (δₖ - ωₖ₋₁ / γₖ₋₁) = pₖᵀ A pₖ.
+    # It is possible to show that σₖ² (δₖ - ωₖ₋₁ / γₖ₋₁) = pₖᴴ A pₖ.
     γ = one(T) / (δ - ω / γ)  # γₖ = 1 / (δₖ - ωₖ₋₁ / γₖ₋₁)
     indefinite |= (γ ≤ 0)
     (check_curvature & indefinite) && continue
@@ -172,7 +172,7 @@ function cg_lanczos!(solver :: CgLanczosSolver{T,FC,S}, A, b :: AbstractVector{F
     end
     @. 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₊₁ᵀ M vₖ₊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
     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.