Improve documentation from PaulaGarciaMolina/main

docs/algorithms/arnoldi.rst

@@ -38,7 +38,7 @@ based on previous estimates, using the formula :math:`|{\xi_{k+1}}\rangle=(1-\ga
 memory factor :math:`\gamma=-0.75`.
 
 
-The rule in :ref:`alg_descent` is a particular case of the Arnoldi iteration with a Krylov basis with  $L=2$.
+The rule in :ref:`alg_descent` is a particular case of the Arnoldi iteration with a Krylov basis with  :math:`L=2`.
 
 Ref. :cite:t:`GarciaMolina2024` presents this algorithm and its implementation for global optimization problems. It is also suitable for evolution problems.
 

docs/algorithms/generated/seemps.evolution.euler.euler.rst

@@ -0,0 +1,11 @@
+
+
+seemps.evolution.euler.euler
+============================
+
+.. currentmodule:: seemps.evolution.euler
+
+
+
+.. autofunction:: seemps.evolution.euler.euler
+

docs/algorithms/generated/seemps.evolution.euler.euler2.rst

@@ -0,0 +1,11 @@
+
+
+seemps.evolution.euler.euler2
+=============================
+
+.. currentmodule:: seemps.evolution.euler
+
+
+
+.. autofunction:: seemps.evolution.euler.euler2
+

docs/algorithms/generated/seemps.evolution.runge_kutta.runge_kutta.rst

@@ -0,0 +1,11 @@
+
+
+seemps.evolution.runge\_kutta.runge\_kutta
+==========================================
+
+.. currentmodule:: seemps.evolution.runge_kutta
+
+
+
+.. autofunction:: seemps.evolution.runge_kutta.runge_kutta
+

docs/algorithms/generated/seemps.evolution.runge_kutta.runge_kutta_fehlberg.rst

@@ -0,0 +1,11 @@
+
+
+seemps.evolution.runge\_kutta.runge\_kutta\_fehlberg
+====================================================
+
+.. currentmodule:: seemps.evolution.runge_kutta
+
+
+
+.. autofunction:: seemps.evolution.runge_kutta.runge_kutta_fehlberg
+

docs/algorithms/index.rst

@@ -9,6 +9,9 @@ Index of algorithms
    tensor_split
    mps_simplification
    gradient_descent
+   cgs
    arnoldi
    dmrg
+   runge_kutta
+   crank_nicolson
    tebd_evolution

docs/algorithms/runge_kutta.rst

@@ -0,0 +1,70 @@
+.. _mps_runge_kutta:
+
+*******************
+Runge-Kutta methods
+*******************
+
+Runge-Kutta methods use a Taylor expansion of the state to approximate
+
+.. math::
+    \psi_{k+1} = \psi_k + \sum_{p}\frac{1}{p!}(-\Delta \beta H)^p\frac{\partial^{(p)} \psi}{\partial \beta^{(p)}}.
+
+:math:`\Delta \beta` can either be real or imaginary time, and hence these
+methods are suitable for both evolution and optimization problems. When using
+MPS, these methods perform a global optimization of the MPS at each step,
+i.e., they update all tensors simultaneously.
+
+The order of the expansion `p` determines the truncation error of the method, which
+is :math:`O(\Delta \beta ^{p+1})`, and also the cost of the method, since
+a higher order implies more operations. Thus, it is important to consider 
+trade-off in cost and accuracy to choose the most suitable method for each application.
+
+The SeeMPS library considers four methods.
+
+1. Euler method
+----------------
+
+This is an explicit, first-order Taylor approximation of the evolution, with an error :math:`\mathcal{O}(\Delta\beta^2)`
+and a simple update with a fixed time-step :math:`\beta_k = k \Delta\beta`.
+
+.. math::
+    \psi_0 &= \psi(\beta_0), \\
+    \psi_{k+1} &= \psi_k - \Delta\beta H \psi_k, \quad \text{for } k=0,1,\dots,N-1.
+
+2. Improved Euler or Heun method
+---------------------------------
+
+This is a second-order, fixed-step explicit method that uses two matrix-vector multiplications and two linear combinations of 
+vectors to achieve an error :math:`\mathcal{O}(\Delta\beta^3)`.
+
+.. math::
+    \psi_{k+1} = \psi_k - \frac{\Delta\beta}{2} \left[v_1 + H(\psi_k - \Delta\beta v_1)\right], \\
+    \text{with } v_1 = H \psi_k.
+
+3. Fourth-order Runge-Kutta method
+-----------------------------------
+
+This algorithm achieves an error :math:`\mathcal{O}(\Delta\beta^5)` using four matrix-vector multiplications and four linear combinations of vectors.
+
+.. math::
+    \psi_{k+1} &= \psi_k + \frac{\Delta\beta}{6}(v_1 + 2v_2 + 2v_3 + v_4), \\
+    v_1 &= -H \psi_k, \\
+    v_2 &= -H\left(\psi_k + \frac{\Delta\beta}{2}v_1\right), \\
+    v_3 &= -H\left(\psi_k + \frac{\Delta\beta}{2}v_2\right), \\
+    v_4 &= -H\left(\psi_k + \Delta\beta v_3\right).
+
+4. Runge-Kutta-Fehlberg method  
+-------------------------------
+The Runge-Kutta-Fehlberg algorithm is an adaptative step-size solver that combines a fifth-order accurate integrator  
+:math:`O(\Delta\beta^5)` with a sixth-order error estimator  :math:`O(\Delta\beta^6)`. This combination dynamically adjusts the step size  :math:`\Delta\beta` 
+to maintain the integration error within a specified tolerance. The method requires an initial estimate of the step size, which can be obtained from a simpler 
+method. Each iteration involves six matrix-vector multiplications and six linear combinations, and it may repeat the evolution steps if the proposed step size 
+is deemed unsuitable.
+
+.. autosummary::
+    :toctree: generated/
+
+    ~seemps.evolution.euler.euler
+    ~seemps.evolution.euler.euler2
+    ~seemps.evolution.runge_kutta.runge_kutta
+    ~seemps.evolution.runge_kutta.runge_kutta_fehlberg

docs/seemps_analysis_differentiation.rst

@@ -40,6 +40,8 @@ leading to
 
 This approximation is improved with a smoother formula to avoid noise resilience following `Holoborodko <http://www.holoborodko.com/pavel/numerical-methods/numerical-derivative/smooth-low-noise-differentiators>`_.
 
+An example on how to use these functions is shown in `Differentiation.ipynb <https://github.com/juanjosegarciaripoll/seemps2/blob/main/examples/Differentiation.ipynb>`_.
+
 .. autosummary::
     :toctree: generated/
 

docs/seemps_examples.rst

@@ -11,6 +11,8 @@ In the directory `examples` of the library, we have created various Jupyter note
 - `Computation of a spin-1/2 model ground state using DMRG <https://github.com/juanjosegarciaripoll/seemps2/blob/main/examples/DMRG.ipynb>`_.
 - `Solution of a Hamiltonian PDE <https://github.com/juanjosegarciaripoll/seemps2/blob/main/examples/PDE.ipynb>`_.
 - `Function interpolation <https://github.com/juanjosegarciaripoll/seemps2/blob/main/examples/Interpolation.ipynb>`_.
+- `Function differentiation <https://github.com/juanjosegarciaripoll/seemps2/blob/main/examples/Differentiation.ipynb>`_.
+
 
 Further examples of use appear in the datasets and associated codes to the following publications: