[documentation] Use Aᴴ instead of Aᵀ

README.md

@@ -71,7 +71,7 @@ Overdetermined sytems are less common but also occur.
 4. Adjoint systems
 
 <p align="center">
-  <b><i>Ax = b</i></b> &nbsp; and &nbsp; <b><i>Aᵀy = c</i></b>
+  <b><i>Ax = b</i></b> &nbsp; and &nbsp; <b><i>Aᴴy = c</i></b>
 </p>
 
 where **_A_** can have any shape.
@@ -81,7 +81,7 @@ where **_A_** can have any shape.
 <p align="center">
   [<b><i>M </i></b>&nbsp;&nbsp;&nbsp;<b><i> A</i></b>]&nbsp; [<b><i>x</i></b>]            =           [<b><i>b</i></b>]
   <br>
-  [<b><i>Aᵀ</i></b>&nbsp;&nbsp;      <b><i>-N</i></b>]&nbsp; [<b><i>y</i></b>]&nbsp;&nbsp;&nbsp;&nbsp;[<b><i>c</i></b>]
+  [<b><i>Aᴴ</i></b>&nbsp;&nbsp;      <b><i>-N</i></b>]&nbsp; [<b><i>y</i></b>]&nbsp;&nbsp;&nbsp;&nbsp;[<b><i>c</i></b>]
 </p>
 
 where **_A_** can have any shape.
@@ -94,7 +94,7 @@ where **_A_** can have any shape.
   [<b><i>B</i></b>&nbsp;&nbsp;&nbsp;<b><i>N</i></b>]&nbsp; [<b><i>y</i></b>]&nbsp;&nbsp;&nbsp;&nbsp;[<b><i>c</i></b>]
 </p>
 
-where **_A_** can have any shape and **_B_** has the shape of **_Aᵀ_**.
+where **_A_** can have any shape and **_B_** has the shape of **_Aᴴ_**.
 **_A_**, **_B_**, **_b_** and **_c_** must be all nonzero.
 
 Krylov solvers are particularly appropriate in situations where such problems must be solved but a factorization is not possible, either because:

docs/src/examples/tricg.md

@@ -14,7 +14,7 @@ N = diagm(0 => [5.0 * i for i = 1:n])
 c = -b
 
 # [I   A] [x] = [b]
-# [Aᵀ -I] [y]   [c]
+# [Aᴴ -I] [y]   [c]
 (x, y, stats) = tricg(A, b, c)
 K = [eye(m) A; A' -eye(n)]
 B = [b; c]
@@ -23,7 +23,7 @@ resid = norm(r)
 @printf("TriCG: Relative residual: %8.1e\n", resid)
 
 # [-I   A] [x] = [b]
-# [ Aᵀ  I] [y]   [c]
+# [ Aᴴ  I] [y]   [c]
 (x, y, stats) = tricg(A, b, c, flip=true)
 K = [-eye(m) A; A' eye(n)]
 B = [b; c]
@@ -32,7 +32,7 @@ resid = norm(r)
 @printf("TriCG: Relative residual: %8.1e\n", resid)
 
 # [I   A] [x] = [b]
-# [Aᵀ  I] [y]   [c]
+# [Aᴴ  I] [y]   [c]
 (x, y, stats) = tricg(A, b, c, spd=true)
 K = [eye(m) A; A' eye(n)]
 B = [b; c]
@@ -41,7 +41,7 @@ resid = norm(r)
 @printf("TriCG: Relative residual: %8.1e\n", resid)
 
 # [-I    A] [x] = [b]
-# [ Aᵀ  -I] [y]   [c]
+# [ Aᴴ  -I] [y]   [c]
 (x, y, stats) = tricg(A, b, c, snd=true)
 K = [-eye(m) A; A' -eye(n)]
 B = [b; c]
@@ -50,7 +50,7 @@ resid = norm(r)
 @printf("TriCG: Relative residual: %8.1e\n", resid)
 
 # [τI    A] [x] = [b]
-# [ Aᵀ  νI] [y]   [c]
+# [ Aᴴ  νI] [y]   [c]
 (τ, ν) = (1e-4, 1e2)
 (x, y, stats) = tricg(A, b, c, τ=τ, ν=ν)
 K = [τ*eye(m) A; A' ν*eye(n)]
@@ -60,7 +60,7 @@ resid = norm(r)
 @printf("TriCG: Relative residual: %8.1e\n", resid)
 
 # [M⁻¹  A  ] [x] = [b]
-# [Aᵀ  -N⁻¹] [y]   [c]
+# [Aᴴ  -N⁻¹] [y]   [c]
 (x, y, stats) = tricg(A, b, c, M=M, N=N, verbose=1)
 K = [inv(M) A; A' -inv(N)]
 H = BlockDiagonalOperator(M, N)

docs/src/examples/trimr.md

@@ -14,7 +14,7 @@ m, n = size(A)
 c = -b
 
 # [D   A] [x] = [b]
-# [Aᵀ  0] [y]   [c]
+# [Aᴴ  0] [y]   [c]
 llt_D = cholesky(D)
 opD⁻¹ = LinearOperator(Float64, 5, 5, true, true, (y, v) -> ldiv!(y, llt_D, v))
 opH⁻¹ = BlockDiagonalOperator(opD⁻¹, eye(n))
@@ -34,7 +34,7 @@ N = diagm(0 => [5.0 * i for i = 1:n])
 c = -b
 
 # [I   A] [x] = [b]
-# [Aᵀ -I] [y]   [c]
+# [Aᴴ -I] [y]   [c]
 (x, y, stats) = trimr(A, b, c)
 K = [eye(m) A; A' -eye(n)]
 B = [b; c]
@@ -43,7 +43,7 @@ resid = norm(r)
 @printf("TriMR: Relative residual: %8.1e\n", resid)
 
 # [M   A] [x] = [b]
-# [Aᵀ -N] [y]   [c]
+# [Aᴴ -N] [y]   [c]
 ldlt_M = ldl(M)
 ldlt_N = ldl(N)
 opM⁻¹ = LinearOperator(Float64, size(M,1), size(M,2), true, true, (y, v) -> ldiv!(y, ldlt_M, v))

docs/src/gpu.md

@@ -50,7 +50,7 @@ using CUDA, CUDA.CUSPARSE
 A_gpu = CuSparseMatrixCSC(A_cpu)  # A = CuSparseMatrixCSR(A_cpu)
 b_gpu = CuVector(b_cpu)
 
-# LLᵀ ≈ A for CuSparseMatrixCSC or CuSparseMatrixCSR matrices
+# LLᴴ ≈ A for CuSparseMatrixCSC or CuSparseMatrixCSR matrices
 P = ic02(A_gpu, 'O')
 
 # Solve Py = x

docs/src/index.md

@@ -46,15 +46,15 @@ Overdetermined sytems are less common but also occur.
 4 - Adjoint systems
 
 ```math
-  Ax = b \quad \text{and} \quad A^T y = c
+  Ax = b \quad \text{and} \quad A^H y = c
 ```
 
 where **_A_** can have any shape.
 
 5 - Saddle-point and symmetric quasi-definite (SQD) systems
 
 ```math
-  \begin{bmatrix} M & \phantom{-}A \\ A^T & -N \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \left(\begin{bmatrix} b \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ c \end{bmatrix},\begin{bmatrix} b \\ c \end{bmatrix}\right)
+  \begin{bmatrix} M & \phantom{-}A \\ A^H & -N \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \left(\begin{bmatrix} b \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ c \end{bmatrix},\begin{bmatrix} b \\ c \end{bmatrix}\right)
 ```
 
 where **_A_** can have any shape.
@@ -65,7 +65,7 @@ where **_A_** can have any shape.
   \begin{bmatrix} M & A \\ B & N \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} b \\ c \end{bmatrix}
 ```
 
-where **_A_** can have any shape and **_B_** has the shape of **_Aᵀ_**.
+where **_A_** can have any shape and **_B_** has the shape of **_Aᴴ_**.
 **_A_**, **_B_**, **_b_** and **_c_** must be all nonzero.
 
 Krylov solvers are particularly appropriate in situations where such problems must be solved but a factorization is not possible, either because:

docs/src/warm_start.md

@@ -41,14 +41,14 @@ Explicit restarts cannot be avoided in certain block methods, such as TriMR, due
 
 ```julia
 # [E  A] [x] = [b]
-# [Aᵀ F] [y]   [c]
+# [Aᴴ F] [y]   [c]
 M = inv(E)
 N = inv(F)
 x₀, y₀, stats = trimr(A, b, c, M=M, N=N)
 
 # E and F are not available inside TriMR
 b₀ = b -  Ex₀ - Ay
-c₀ = c - Aᵀx₀ - Fy
+c₀ = c - Aᴴx₀ - Fy
 
 Δx, Δy, stats = trimr(A, b₀, c₀, M=M, N=N)
 x = x₀ + Δx