Add comments to tutorial

docs/src/tutorials/qless.md

@@ -1,6 +1,7 @@
 ```@example qless1
 using QRMumps, LinearAlgebra, SparseArrays, Printf
 
+# Initialize data
 m, n = 5, 7
 irn = [1, 3, 5, 2, 3, 5, 1, 4, 4, 5, 2, 1, 3]
 jcn = [1, 1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7]
@@ -12,18 +13,27 @@ y_star = [0.0, 1.0, 2.0, 3.0, 4.0]
 y₁ = zeros(n)
 y₂ = zeros(m)
 
+# Initialize QRMumps
 qrm_init()
 
+# Initialize data structures
 spmat = qrm_spmat_init(A)
 spfct = qrm_spfct_init(spmat)
 
+# Specify not storing Q for the QR factorization
 qrm_set(spfct, "qrm_keeph", 0)
+
+# Perform symbolic analysis of Aᵀ and factorize Aᵀ = QR
 qrm_analyse!(spmat, spfct; transp='t')
 qrm_factorize!(spmat, spfct, transp='t')
 
+# Solve Rᵀy₁ =  b  
 qrm_solve!(spfct, b, y₁; transp='t')
+
+# Solve Ry₂ = y₁
 qrm_solve!(spfct, y₁, y₂; transp='n')
 
+# Overall, RᵀRy₂ = b. Equivalently, RᵀQᵀQRy₂ = b or AAᵀy₂ = b
 error_norm = norm(y₂ - y_star)
 
 @printf("Error norm ‖y* - y‖ = %10.5e\n", error_norm)
@@ -32,6 +42,7 @@ error_norm = norm(y₂ - y_star)
 ```@example qless2
 using QRMumps, LinearAlgebra, SparseArrays, Printf
 
+# Initialize data
 m, n = 7, 5
 irn = [1, 1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7]
 jcn = [1, 3, 5, 2, 3, 5, 1, 4, 4, 5, 2, 1, 3]
@@ -41,20 +52,29 @@ A = sparse(irn, jcn, val, m, n)
 b = [22.0, 2.0, 13.0, 8.0, 17.0, 6.0, 5.0]
 x_star = [1.0, 2.0, 3.0, 4.0, 5.0]
 
+# Initialize QRMumps
 qrm_init()
 
+# Initialize data structures
 spmat = qrm_spmat_init(A)
 spfct = qrm_spfct_init(spmat)
 
+# Specify not storing Q for the QR factorization
 qrm_set(spfct, "qrm_keeph", 0)
 
+# Perform symbolic analysis of A and factorize A = QR
 qrm_analyse!(spmat, spfct)
 qrm_factorize!(spmat, spfct)
 
 z = A'*b
+
+# Solve Rᵀx₁ = z = Aᵀb
 x₁ = qrm_solve(spfct, z; transp = 't')
+
+# Solve Rx₂ = x₁ 
 x₂ = qrm_solve(spfct, x₁; transp = 'n')
 
+# Overall, RᵀRx₂ = Aᵀb. Equivalently, RᵀQᵀQRx₂ = Aᵀb or AᵀAx₂ = Aᵀb
 error_norm = norm(x₂ - x_star)
 
 @printf("Error norm ‖x* - x‖ = %10.5e\n", error_norm)