Fix typos in doc

ChainOhmT/gauss.jl

@@ -1,16 +1,11 @@
 using LinearAlgebra
 
-""" GAUSS Gauss quadrature rule.
+""" 
+    gauss(N,ab)
 
-    Given a weight function w encoded by the nx2 array ab of the
-    first n recurrence coefficients for the associated orthogonal
-    polynomials, the first column of ab containing the n alpha-
-    coefficients and the second column the n beta-coefficients,
-    the call xw = GAUSS(n,ab) generates the nodes and weights xw of
-    the n-point Gauss quadrature rule for the weight function w.
-    The nodes, in increasing order, are stored in the first
-    column, the n corresponding weights in the second column, of
-    the nx2 array xw.
+Gauss quadrature rule for `N` sites on an interval `ab`. Given a weight function w encoded by the `N`x2 array `ab` of the first `N` recurrence coefficients for the associated orthogonal
+polynomials, the first column of `ab` containing the `N` alpha-coefficients and the second column the `N` beta-coefficients, the call xw = gauss(N,ab) generates the nodes and weights xw of the `N`-point Gauss quadrature rule for the weight function w.
+The nodes, in increasing order, are stored in the first column, the n corresponding weights in the second column, of the `N`x2 array xw.
 """
 function gauss(N,ab)
     N0 = size(ab,1)

ChainOhmT/lanczos.jl

@@ -1,17 +1,9 @@
-""" LANCZOS Lanczos algorithm.
+""" 
+    lanczos(N,xw)
 
-    Given the discrete inner product whose nodes are contained
-    in the first column, and whose weights are contained in the
-    second column, of the nx2 array xw, the call ab=LANCZOS(n,xw)
-    generates the first n recurrence coefficients ab of the
-    corresponding discrete orthogonal polynomials. The n alpha-
-    coefficients are stored in the first column, the n beta-
-    coefficients in the second column, of the nx2 array ab.
+Given the discrete inner product whose nodes are contained in the first column, and whose weights are contained in the second column, of the `N`x2 array `xw`, the call ab=lanczos(N,xw) generates the first `N` recurrence coefficients ab of the corresponding discrete orthogonal polynomials. The `N` alpha-coefficients are stored in the first column, the `N` beta-coefficients in the second column, of the `N`x2 array ab.
 
-    The script is adapted from the routine RKPW in
-    W.B. Gragg and W.J. Harrod, ``The numerically stable
-    reconstruction of Jacobi matrices from spectral data'',
-    Numer. Math. 44 (1984), 317-335.
+The script is adapted from the routine RKPW in W.B. Gragg and W.J. Harrod, ``The numerically stable reconstruction of Jacobi matrices from spectral data'', Numer. Math. 44 (1984), 317-335.
 """
 function lanczos(N,xw) #return ab
     Ncap = size(xw,1)

ChainOhmT/stieltjes.jl

@@ -1,12 +1,7 @@
-""" STIELTJES Discretized Stieltjes procedure.
+""" 
+    stieltjes(N,xw)
 
-    Given the discrete inner product whose nodes are contained
-    in the first column, and whose weights are contained in the
-    second column, of the nx2 array xw, the call ab=STIELTJES(n,xw)
-    generates the first n recurrence coefficients ab of the
-    corresponding discrete orthogonal polynomials. The n alpha-
-    coefficients are stored in the first column, the n beta-
-    coefficients in the second column, of the nx2 array ab.
+Discretized Stieltjes procedure. Given the discrete inner product whose nodes are contained in the first column, and whose weights are contained in the second column, of the `N`x2 array `xw`, the call ab=stieltjes(N,xw) generates the first `N` recurrence coefficients ab of the corresponding discrete orthogonal polynomials. The `N` alpha- coefficients are stored in the first column, the `N` beta-coefficients in the second column, of the `N`x2 array ab.
 """
 function stieltjes(N,xw) #return ab
     #tiny = 10*realmin

src/fundamentals.jl

@@ -392,7 +392,7 @@ end
 """
     dynamap(ps1,ps2,ps3,ps4)
 
-Calulate complete dynamical map to time step at which ps1, ps2, ps3 and ps4 are specified.
+Calculate complete dynamical map to time step at which ps1, ps2, ps3 and ps4 are specified.
 
 """
 function dynamap(ps1,ps2,ps3,ps4)

src/models.jl

@@ -457,7 +457,7 @@ The spin is on site ``N_l + 1`` of the MPS, surrounded by the left chain modes a
 This Hamiltonain is unitarily equivalent (before the truncation to `N` sites) to the spin-boson Hamiltonian defined by
 
 ``
-H =  \\frac{ω_0}{2}σ_z + Δσ_x + σ_x\\int_0^∞ dω\\sqrt{\\frac{J(ω)}{π}}(b_ω^\\dagger+b_ω) + \\int_0^∞ dω ωb_ω^\\dagger b_ωi + σ_x\\int_0^∞ dω\\sqrt{\\frac{J^l(ω)}{π}}(d_ω^\\dagger+d_ω) + \\int_0^∞ dω ωd_ω^\\dagger d_ω
+H =  \\frac{ω_0}{2}σ_z + Δσ_x + σ_x\\int_0^∞ dω\\sqrt{J(ω)}(b_ω^\\dagger+b_ω) + \\int_0^∞ dω ωb_ω^\\dagger b_ωi + σ_x\\int_0^∞ dω\\sqrt{J^l(ω)}(d_ω^\\dagger+d_ω) + \\int_0^∞ dω ωd_ω^\\dagger d_ω
 ``.
 
 The chain parameters, supplied by `chainparams`=``[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``, can be chosen to represent any arbitrary spectral density ``J(ω)`` at any temperature. The two chains can have a different spectral density.
@@ -582,7 +582,7 @@ The spin is on site 1 of the MPS and the bath modes are to the right.
 This Hamiltonain is unitarily equivalent (before the truncation to `N` sites) to the spin-boson Hamiltonian defined by
 
 ``
-H =  \\frac{ω_0}{2}σ_z + σ_z\\int_0^∞ dω\\sqrt{\\frac{J(ω)}{π}}(b_ω^\\dagger+b_ω) + \\int_0^∞ dω ωb_ω^\\dagger b_ω
+H =  \\frac{ω_0}{2}σ_z + σ_z\\int_0^∞ dω\\sqrt{J(ω)}(b_ω^\\dagger+b_ω) + \\int_0^∞ dω ωb_ω^\\dagger b_ω
 ``.
 
 The chain parameters, supplied by `chainparams`=``[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``, can be chosen to represent any arbitrary spectral density ``J(ω)`` at any temperature.
@@ -708,12 +708,12 @@ end
 """
     puredephasingmpo(ΔE, dchain, Nchain, chainparams; tree=false)
 
-    Generate MPO for a pure dephasing model, defined by the Hamiltonian
-    ``H = \\frac{ΔE}{2} σ_z +  \\frac{σ_z}{2} c_0 (b_0^\\dagger + b_0) + \\sum_{i=0}^{N-1} t_i (b_{i+1}^\\dagger b_i +h.c.) + \\sum_{i=0}^{N-1} ϵ_i b_i^\\dagger b_i  ``
+Generate MPO for a pure dephasing model, defined by the Hamiltonian
+``H = \\frac{ΔE}{2} σ_z +  \\frac{σ_z}{2} c_0 (b_0^\\dagger + b_0) + \\sum_{i=0}^{N-1} t_i (b_{i+1}^\\dagger b_i +h.c.) + \\sum_{i=0}^{N-1} ϵ_i b_i^\\dagger b_i  ``
 
-    The spin is on site 1 of the MPS and the bath modes are to the right.
+The spin is on site 1 of the MPS and the bath modes are to the right.
 
-    ### Arguments
+### Arguments
     * `ΔE::Real`: energy splitting of the spin
     * `dchain::Int`: physical dimension of the chain sites truncated Hilbert spaces
     * `Nchain::Int`: number of sites in the chain
@@ -737,22 +737,18 @@ end
 """
     tightbinding_mpo(N, ϵd, chainparams1, chainparams2)
 
-    Generate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well) 
-    of energy ϵd at the center. The impurity is coupled to two leads, each described by a set of chain parameters.
-    The interactions are nearest-neighbour, with the first N/2-1 sites corresponding to the first lead,
-    the Nth site corresponding to the impurity, and the rest of the sites corresponding to the second
-    lead.
+Generate MPO for a tight-binding chain of N fermionic sites with a single impurity site (fermionic as well) 
+of energy ϵd at the center. The impurity is coupled to two leads, each described by a set of chain parameters.
+The interactions are nearest-neighbour, with the first N/2-1 sites corresponding to the first lead, the Nth site corresponding to the impurity, and the rest of the sites corresponding to the second lead.
 
-    # Arguments
+# Arguments
 
     * `N::Int`: number of sites in the chain
     * `ϵd::Real`: energy of the impurity site at the center, as Ed - μ, where μ is the chemical potential
     * chainparams1::Array{Real,1}: chain parameters for the first lead
     * chainparams2::Array{Real,1}: chain parameters for the second lead
 
-    The chain parameters are given in the standard form: `chainparams` ``=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``.
-
-
+The chain parameters are given in the standard form: `chainparams` ``=[[ϵ_0,ϵ_1,...],[t_0,t_1,...],c_0]``.
 """
 function tightbinding_mpo(N, ϵd, chainparams1, chainparams2)
 
@@ -996,7 +992,7 @@ end
 Generate a MPO for a one-dimensional bosonic bath spatially correlated to a multi-component system 
 
 ``
-H_B + H_int = \\int_{-∞}^{+∞} dk ω_k b_k^\\dagger b_k + ∑_j \\int_{-∞}^{+∞}dk \\sqrt{J(k)}(A_j b_k e^{i k R_j} + h.c.)
+H_B + H_{int} = \\int_{-∞}^{+∞} dk ω_k b_k^\\dagger b_k + ∑_j \\int_{-∞}^{+∞}dk \\sqrt{J(k)}(A_j b_k e^{i k R_j} + h.c.)
 ``.
 
 The interactions between the system and the chain-mapped bath are long range, i.e. each site interacts with all the chain modes. The spectral density is assumed to be Ohmic ``J(ω) = 2αωc(ω/ωc)^s``.
@@ -1341,10 +1337,10 @@ end
 Generate a MPO for a system described in space with a reaction coordinate (RC) tensor. The RC tensor is coupled to a bosonic bath, taking into account the induced reorganization energy. 
 
 ``
-H_S + H_RC + H_int^{S-RC} = \\omega^0_{e} |e\\rangle \\langle e| + \\omega^0_{k} |k\\rangle \\langle k| + \\Delta (|e\\rangle \\langle k| + |k\\rangle \\langle e|) + \\omega_{RC} (d^{\\dagger}d + \\frac{1}{2}) + g_{e} |e\\rangle \\langle e|( d + d^{\\dagger})+ g_{k} |k \\rangle \\langle k|( d + d^{\\dagger})
+H_S + H_{RC} + H_{int}^{S-RC} = \\omega^0_{e} |e\\rangle \\langle e| + \\omega^0_{k} |k\\rangle \\langle k| + \\Delta (|e\\rangle \\langle k| + |k\\rangle \\langle e|) + \\omega_{RC} (d^{\\dagger}d + \\frac{1}{2}) + g_{e} |e\\rangle \\langle e|( d + d^{\\dagger})+ g_{k} |k \\rangle \\langle k|( d + d^{\\dagger})
 ``
 ``
-H_B + H_int^{RC-B} = \\int_{-∞}^{+∞} dk ω_k b_k^\\dagger b_k - (d + d^{\\dagger})\\int_0^∞ dω\\sqrt{J(ω)}(b_ω^\\dagger+b_ω) + \\lambda_{reorg}(d + d^{\\dagger})^2
+H_B + H_{int}^{RC-B} = \\int_{-∞}^{+∞} dk ω_k b_k^\\dagger b_k - (d + d^{\\dagger})\\int_0^∞ dω\\sqrt{J(ω)}(b_ω^\\dagger+b_ω) + \\lambda_{reorg}(d + d^{\\dagger})^2
 ``.
 ``
 \\lambda_{reorg} = \\int \\frac{J(\\omega)}{\\omega}d\\omega
@@ -1392,4 +1388,4 @@ function protontransfermpo(ω0e,ω0k,x0e,x0k, Δ, dRC, d, N, chainparams, RCpara
     chain = hbathchain(N, d, chainparams; coupletox=true)
 
     return Any[M,MRC,chain...]
-end
+end

src/mpsBasics.jl

@@ -542,7 +542,7 @@ apply1siteoperator!(A, O, site::Int) = apply1siteoperator!(A, O, [site])
 """
     applympo!(A, H)
 
-Apply an MPO H on the MPS A. H must have the same number of site than A. The resulting MPS A is the MPS modified by the mpo H.
+Apply an MPO H on the MPS A. H must have the same number of site than A. The resulting MPS A is the MPS modified by the MPO H.
 
 """
 function applympo!(A, H)
@@ -729,7 +729,7 @@ end
 """
     displacedchainmps(A::Vector{Any}, N::Int, Nm::Int, γ::Any)
 
-    Given a mps A, return a mps B where the `Nm`-long chain is displaced by `γ` without displacing the `N`-long system.
+Given a MPS A, return a MPS B where the `Nm`-long chain is displaced by `γ` without displacing the `N`-long system.
 """
 function displacedchainmps(A::Vector{Any}, N::Int, Nm::Int, γ::Any)