#!/usr/bin/env python
# -*- coding: utf-8 -*-
r"""
Matrices
========
There is a large number of Matrix types that are all derived from the
base class ``MatrixBase``. They do not have to store elements but can be
logical or wrappers. They just have to provide the functions
``A.cols()``, ``A.rows()`` (column and row numbers),
``A.mult(x)``\ :math:`=A\cdot x` and ``A.transMult(y)``\ :math:`=A^T\cdot y`.
"""
# sphinx_gallery_thumbnail_number = 3
# We start off with the typical imports
import numpy as np
import pygimli as pg
# %%%
# Dense matrix ``Matrix``
# -----------------------
#
# All elements are stored column-wise, i.e. all rows ``A[i]`` are of type
# ``pg.Vector``. This matrix is used for storing dense data (like ERT
# Jacobians) and doing simple algebra.
#
A = pg.Matrix(3, 4)
A[0, 0] = 1
A[2, 2] = -1
A[1] = np.arange(4)
print(A)
x = np.arange(4) + 5
A*x
# %%%
# Exists also as complex matrix under ``pg.matrix.CMatrix``.
#
# %%%
# Index-based sparse matrix ``SparseMapMatrix``
# ---------------------------------------------
#
# Sparse matrix (most elements are zero) with single access and. Typical
# for traveltime Jacobian (only certain cells covered by individual rays)
# or constraint matrices.
#
A = pg.SparseMapMatrix(2, 3)
A.setVal(0, 0, -1)
A.setVal(0, 1, +1) # first-order derivative
A.setVal(1, 0, -1)
A.setVal(1, 1, +2)
A.setVal(1, 2, -1) # second-order derivative
x = [10, 12, 13]
print(A * x)
ax, _ = pg.show(A)
# %%%
# Exists also as complex-valued variant ``pg.matrix.CSparseMapMatrix``.
#
# %%%
# Column-compressed matrix ``SparseMatrix``
# -----------------------------------------
#
# Used for numerical approximation of partial differential equations like
# finite-element or finite volume. Not typically used unless efficiency is
# of importance. It exists also complex-valued as ``pg.matrix.CSparseMatrix``.
#
# %%%
# Diagonal matrices
# -----------------
#
# First, there is an identity matrix ``IdentityMatrix``. No elements
# stored at all. Important for constraint matrices when combined into
# ``BlockMatrix`` (see below). More generally, there is a diagonal matrix
# ``DiagonalMatrix`` where only its diagonal is stored as vector.
#
A = pg.matrix.IdentityMatrix(3) # , 2.0)
x = [10, 11, 12]
A*x
# %%%
# Weighted matrices
# -----------------
# ``MultLeftMatrix``/``MultRightMatrix``/``MultLeftRightMatrix``
#
# Often, matrices are weighted from either side by a vector, e.g. data
# error weighting of the Jacobian matrix, data and model transformations,
# or weighting individual smoothness parts according to the roughness so
# that only the weighting is changed and not the matrix.
#
A = pg.Matrix(3, 4)
A += 1
w = [2, 3, -1]
B = pg.matrix.MultLeftMatrix(A, w)
x = np.arange(4)
print(A*x)
print(B*x)
# %%%
# Combinations of matrices
# ------------------------
#
# Logical matrices can combine different other matrices (of arbitrary
# type) avoiding double memory storage by multiplication (``Mult2Matrix``)
# or addition (``Add2Matrix``).
#
A = pg.Matrix(2, 3)
A += 1
B = pg.SparseMapMatrix(3, 4)
C = pg.matrix.Mult2Matrix(A, B)
x = np.arange(4)
C * x
# %%%
# Block matrices
# --------------
#
# The most important type is the ``BlockMatrix``, where arbitrary matrices
# are combined into a logical matrix. This is of importance for inversion
# frameworks: \* joint inversion: Jacobian matrices are concatenated \*
# combination of different constrains: combining different regularization
# \* laterally, spatially or temporally constrained inversion:
# regularization between model cells of each frame but also between the
# frames Note that the matrices only have to be defined once and can
# appear multiply.
#
A = pg.BlockMatrix()
A1 = pg.Matrix(2, 2)
A.addMatrix(A1, 0, 0)
A.addMatrix(A1, 4, 0, scale=2.0)
A2 = pg.matrix.IdentityMatrix(5)
A.addMatrix(A2, 1, 2)
A3 = pg.SparseMapMatrix(2, 3)
A.addMatrix(A3, 2, 3)
print(A)
ax, _ = pg.show(A)
# %%%
# Matrix combinations
# -------------------
# There are also simpler types of matrix combinations: \*
# ``H2Matrix``/``V2Matrix``: two matrices below/next to each other \*
# ``HNMatrix``/``VNMatrix``: one matrix repeated N times
# horizontally/vertically \* ``NDMatrix``: block diagonal matrix
#
# %%%
# Matrix wrappers
# ---------------
# TransposedMatrix avoids transposing any matrix by exchanging left/right mult.
# SquaredMatrix keeps only the matrix A but works as A^T @ A
# SquaredTransposeMatrix keeps only the matrix A but works as A @ A.T
# RealNumpyMatrix holds a real-valued numpy array
# ComplexNumpyMatrix holds a complex-valued numpy array in a real pg matrix
# NumpyMatrix
#
# %%%
# Matrix generators
# -----------------
# Often, several matrices or even the same one have to be combined.
# RepeatHMatrix, RepeatVMatrix, RepeatDMatrix hold a single matrix that is
# repeated horizontally, vertically or diagonally.
# NDMatrix,
# FrameConstraintMatrix is a special generator for constraining cells of every
# (e.g. timelapse) frame and moreover the frames with each other.
F = pg.matrix.FrameConstraintMatrix(A3, 3)
ax, _ = pg.show(F)
print(F)
# %%%
# Geostatistical constraint matrix
# --------------------------------
#
# For geostatistical constraints, a correlation matrix is computed using
# correlation lengths and angles to define their directions. To access its
# inverse root in a way that avoids matrix inversion, an eigenvalue
# decomposition is done and the eigenvalues :math:`D`` and -vectors
# :math:`Q` are stored so that the operator
#
# .. math:: C^{-0.5}\cdot x=Q\cdot D^{-0.5}\cdot Q^T \cdot x
#
# Jordi, C., Doetsch, J., Günther, T., Schmelzbach, C. & Robertsson,
# J.O.A. (2018): Geostatistical regularisation operators for geophysical
# inverse problems on irregular meshes. Geophysical Journal International
# 213, 1374- 1386, doi:10.1093/gji/ggy055.
#