.. index:: single: linear_algebra_protocol
.. _linear_algebra_protocol/0:

.. rst-class:: right

**protocol**

``linear_algebra_protocol``
===========================

Linear algebra protocol.

| **Availability:** 
|    ``logtalk_load(linear_algebra(loader))``

| **Author:** Paulo Moura
| **Version:** 1:0:0
| **Date:** 2026-05-05

| **Compilation flags:**
|    ``static``


| **Dependencies:**
|   (none)


| **Remarks:**
|    (none)

| **Inherited public predicates:**
|    (none)

.. contents::
   :local:
   :backlinks: top

Public predicates
-----------------

.. index:: new_vector/3
.. _linear_algebra_protocol/0::new_vector/3:

``new_vector/3``
^^^^^^^^^^^^^^^^

Constructs a numeric vector with the requested length, filled with the given value.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``new_vector(Count,Value,Vector)``
| **Mode and number of proofs:**
|    ``new_vector(+integer,+number,-list(number))`` - ``one``

| **Exceptions:**
|    ``Count`` is a variable:
|        ``instantiation_error``
|    ``Value`` is a variable:
|        ``instantiation_error``
|    ``Count`` is neither a variable nor an integer:
|        ``type_error(integer,Count)``
|    ``Value`` is neither a variable nor a number:
|        ``type_error(number,Value)``
|    ``Count`` is an integer that is less than zero:
|        ``domain_error(non_negative_integer,Count)``


------------

.. index:: new_vector_like/2
.. _linear_algebra_protocol/0::new_vector_like/2:

``new_vector_like/2``
^^^^^^^^^^^^^^^^^^^^^

Constructs a numeric zero vector matching the length of the reference list.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``new_vector_like(Reference,Zeroes)``
| **Mode and number of proofs:**
|    ``new_vector_like(+list,-list(number))`` - ``one``


------------

.. index:: add_vectors/3
.. _linear_algebra_protocol/0::add_vectors/3:

``add_vectors/3``
^^^^^^^^^^^^^^^^^

Adds two numeric vectors element-wise.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``add_vectors(Vector1,Vector2,Vector)``
| **Mode and number of proofs:**
|    ``add_vectors(+list(number),+list(number),-list(number))`` - ``one``


------------

.. index:: subtract_vectors/3
.. _linear_algebra_protocol/0::subtract_vectors/3:

``subtract_vectors/3``
^^^^^^^^^^^^^^^^^^^^^^

Subtracts the second numeric vector from the first element-wise.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``subtract_vectors(Vector1,Vector2,Vector)``
| **Mode and number of proofs:**
|    ``subtract_vectors(+list(number),+list(number),-list(number))`` - ``one``


------------

.. index:: add_scaled_vector/4
.. _linear_algebra_protocol/0::add_scaled_vector/4:

``add_scaled_vector/4``
^^^^^^^^^^^^^^^^^^^^^^^

Adds a scaled numeric vector to another numeric vector.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``add_scaled_vector(Vector,Scale,Vector0,UpdatedVector)``
| **Mode and number of proofs:**
|    ``add_scaled_vector(+list(number),+number,+list(number),-list(number))`` - ``one``

| **Exceptions:**
|    ``Scale`` is a variable:
|        ``instantiation_error``
|    ``Scale`` is neither a variable nor a number:
|        ``type_error(number,Scale)``


------------

.. index:: scale_vector/3
.. _linear_algebra_protocol/0::scale_vector/3:

``scale_vector/3``
^^^^^^^^^^^^^^^^^^

Scales a numeric vector by the given factor.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``scale_vector(Vector,Scale,ScaledVector)``
| **Mode and number of proofs:**
|    ``scale_vector(+list(number),+number,-list(number))`` - ``one``

| **Exceptions:**
|    ``Scale`` is a variable:
|        ``instantiation_error``
|    ``Scale`` is neither a variable nor a number:
|        ``type_error(number,Scale)``


------------

.. index:: dot_product/3
.. _linear_algebra_protocol/0::dot_product/3:

``dot_product/3``
^^^^^^^^^^^^^^^^^

Computes the dot product of two numeric vectors.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``dot_product(Vector1,Vector2,Product)``
| **Mode and number of proofs:**
|    ``dot_product(+list(number),+list(number),-number)`` - ``one``


------------

.. index:: euclidean_norm/2
.. _linear_algebra_protocol/0::euclidean_norm/2:

``euclidean_norm/2``
^^^^^^^^^^^^^^^^^^^^

Computes the Euclidean norm of a numeric vector.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``euclidean_norm(Vector,Norm)``
| **Mode and number of proofs:**
|    ``euclidean_norm(+list(number),-float)`` - ``one``


------------

.. index:: vector_norm/3
.. _linear_algebra_protocol/0::vector_norm/3:

``vector_norm/3``
^^^^^^^^^^^^^^^^^

Computes the norm of a numeric vector for a positive numeric order or the atoms `inf` and `infinity`.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``vector_norm(Vector,Order,Norm)``
| **Mode and number of proofs:**
|    ``vector_norm(+list(number),@term,-number)`` - ``one_or_error``

| **Exceptions:**
|    ``Order`` is a variable:
|        ``instantiation_error``
|    ``Order`` is neither a variable, a number, nor one of the atoms ``inf`` and ``infinity``:
|        ``type_error(number,Order)``
|    ``Order`` is a number but not a valid positive_number_or_infinity:
|        ``domain_error(positive_number_or_infinity,Order)``


------------

.. index:: basis_vector/3
.. _linear_algebra_protocol/0::basis_vector/3:

``basis_vector/3``
^^^^^^^^^^^^^^^^^^

Constructs a numeric basis vector with value 1.0 at the given one-based index and 0.0 elsewhere.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``basis_vector(Size,Index,Vector)``
| **Mode and number of proofs:**
|    ``basis_vector(+integer,+integer,-list(number))`` - ``one``

| **Exceptions:**
|    ``Size`` is a variable:
|        ``instantiation_error``
|    ``Index`` is a variable:
|        ``instantiation_error``
|    ``Size`` is neither a variable nor an integer:
|        ``type_error(integer,Size)``
|    ``Index`` is neither a variable nor an integer:
|        ``type_error(integer,Index)``
|    ``Size`` is an integer that is less than one:
|        ``domain_error(positive_integer,Size)``
|    ``Index`` is an integer that is less than one:
|        ``domain_error(positive_integer,Index)``


------------

.. index:: new_matrix/4
.. _linear_algebra_protocol/0::new_matrix/4:

``new_matrix/4``
^^^^^^^^^^^^^^^^

Constructs a numeric matrix with the requested row and column counts, filled with the given value.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``new_matrix(Rows,Columns,Value,Matrix)``
| **Mode and number of proofs:**
|    ``new_matrix(+integer,+integer,+number,-list(list(number)))`` - ``one``

| **Exceptions:**
|    ``Rows`` is a variable:
|        ``instantiation_error``
|    ``Columns`` is a variable:
|        ``instantiation_error``
|    ``Value`` is a variable:
|        ``instantiation_error``
|    ``Rows`` is neither a variable nor an integer:
|        ``type_error(integer,Rows)``
|    ``Columns`` is neither a variable nor an integer:
|        ``type_error(integer,Columns)``
|    ``Value`` is neither a variable nor a number:
|        ``type_error(number,Value)``
|    ``Rows`` is an integer that is less than zero:
|        ``domain_error(non_negative_integer,Rows)``
|    ``Columns`` is an integer that is less than zero:
|        ``domain_error(non_negative_integer,Columns)``


------------

.. index:: identity_matrix/2
.. _linear_algebra_protocol/0::identity_matrix/2:

``identity_matrix/2``
^^^^^^^^^^^^^^^^^^^^^

Constructs a square identity matrix with the requested size.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``identity_matrix(Size,Matrix)``
| **Mode and number of proofs:**
|    ``identity_matrix(+integer,-list(list(number)))`` - ``one``

| **Exceptions:**
|    ``Size`` is a variable:
|        ``instantiation_error``
|    ``Size`` is neither a variable nor an integer:
|        ``type_error(integer,Size)``
|    ``Size`` is an integer that is less than zero:
|        ``domain_error(non_negative_integer,Size)``


------------

.. index:: matrix_vector_product/3
.. _linear_algebra_protocol/0::matrix_vector_product/3:

``matrix_vector_product/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Computes the matrix-vector product for a numeric matrix and vector, returning the list of row-wise dot products between each matrix row and the vector.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_vector_product(Matrix,Vector,Product)``
| **Mode and number of proofs:**
|    ``matrix_vector_product(+list(list(number)),+list(number),-list(number))`` - ``one``


------------

.. index:: matrix_matrix_product/3
.. _linear_algebra_protocol/0::matrix_matrix_product/3:

``matrix_matrix_product/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Computes the matrix-matrix product for two numeric matrices.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_matrix_product(Matrix1,Matrix2,Product)``
| **Mode and number of proofs:**
|    ``matrix_matrix_product(+list(list(number)),+list(list(number)),-list(list(number)))`` - ``one``


------------

.. index:: gram_matrix/2
.. _linear_algebra_protocol/0::gram_matrix/2:

``gram_matrix/2``
^^^^^^^^^^^^^^^^^

Computes the row Gram matrix of a numeric row matrix by taking all row-wise dot products.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``gram_matrix(Rows,GramMatrix)``
| **Mode and number of proofs:**
|    ``gram_matrix(+list(list(number)),-list(list(number)))`` - ``one``


------------

.. index:: matrix_row_means/2
.. _linear_algebra_protocol/0::matrix_row_means/2:

``matrix_row_means/2``
^^^^^^^^^^^^^^^^^^^^^^

Computes the arithmetic mean of each row in a numeric matrix.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_row_means(Matrix,Means)``
| **Mode and number of proofs:**
|    ``matrix_row_means(+list(list(number)),-list(number))`` - ``one``

| **Exceptions:**
|    ``Matrix`` contains an empty row ``Row``:
|        ``domain_error(minimum_number_of_values(1),Row)``


------------

.. index:: matrix_column_means/2
.. _linear_algebra_protocol/0::matrix_column_means/2:

``matrix_column_means/2``
^^^^^^^^^^^^^^^^^^^^^^^^^

Computes the arithmetic mean of each column in a numeric matrix.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_column_means(Matrix,Means)``
| **Mode and number of proofs:**
|    ``matrix_column_means(+list(list(number)),-list(number))`` - ``one``


------------

.. index:: center_gram_matrix/4
.. _linear_algebra_protocol/0::center_gram_matrix/4:

``center_gram_matrix/4``
^^^^^^^^^^^^^^^^^^^^^^^^

Double-centers a numeric Gram matrix and also returns its row means and total mean.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``center_gram_matrix(GramMatrix,CenteredGramMatrix,RowMeans,TotalMean)``
| **Mode and number of proofs:**
|    ``center_gram_matrix(+list(list(number)),-list(list(number)),-list(number),-number)`` - ``one``

| **Exceptions:**
|    ``GramMatrix`` is empty:
|        ``domain_error(minimum_number_of_values(1),GramMatrix)``
|    ``GramMatrix`` contains an empty row ``Row``:
|        ``domain_error(minimum_number_of_values(1),Row)``


------------

.. index:: center_gram_vector/4
.. _linear_algebra_protocol/0::center_gram_vector/4:

``center_gram_vector/4``
^^^^^^^^^^^^^^^^^^^^^^^^

Centers a numeric Gram vector using training row means and total mean.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``center_gram_vector(GramVector,RowMeans,TotalMean,CenteredGramVector)``
| **Mode and number of proofs:**
|    ``center_gram_vector(+list(number),+list(number),+number,-list(number))`` - ``one``

| **Exceptions:**
|    ``GramVector`` is empty:
|        ``domain_error(minimum_number_of_values(1),GramVector)``


------------

.. index:: outer_product/3
.. _linear_algebra_protocol/0::outer_product/3:

``outer_product/3``
^^^^^^^^^^^^^^^^^^^

Computes the outer product of two numeric vectors.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``outer_product(Vector1,Vector2,Matrix)``
| **Mode and number of proofs:**
|    ``outer_product(+list(number),+list(number),-list(list(number)))`` - ``one``


------------

.. index:: add_matrices/3
.. _linear_algebra_protocol/0::add_matrices/3:

``add_matrices/3``
^^^^^^^^^^^^^^^^^^

Adds two numeric matrices element-wise.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``add_matrices(Matrix1,Matrix2,Matrix)``
| **Mode and number of proofs:**
|    ``add_matrices(+list(list(number)),+list(list(number)),-list(list(number)))`` - ``one``


------------

.. index:: subtract_matrices/3
.. _linear_algebra_protocol/0::subtract_matrices/3:

``subtract_matrices/3``
^^^^^^^^^^^^^^^^^^^^^^^

Subtracts the second numeric matrix from the first element-wise.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``subtract_matrices(Matrix1,Matrix2,Matrix)``
| **Mode and number of proofs:**
|    ``subtract_matrices(+list(list(number)),+list(list(number)),-list(list(number)))`` - ``one``


------------

.. index:: scale_matrix/3
.. _linear_algebra_protocol/0::scale_matrix/3:

``scale_matrix/3``
^^^^^^^^^^^^^^^^^^

Scales each element of a numeric matrix by the given factor.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``scale_matrix(Matrix,Scale,ScaledMatrix)``
| **Mode and number of proofs:**
|    ``scale_matrix(+list(list(number)),+number,-list(list(number)))`` - ``one``


------------

.. index:: frobenius_norm/2
.. _linear_algebra_protocol/0::frobenius_norm/2:

``frobenius_norm/2``
^^^^^^^^^^^^^^^^^^^^

Computes the Frobenius norm of a numeric matrix.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``frobenius_norm(Matrix,Norm)``
| **Mode and number of proofs:**
|    ``frobenius_norm(+list(list(number)),-float)`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: transpose_matrix/2
.. _linear_algebra_protocol/0::transpose_matrix/2:

``transpose_matrix/2``
^^^^^^^^^^^^^^^^^^^^^^

Transposes a numeric matrix represented as a list of row lists.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``transpose_matrix(Matrix,Transpose)``
| **Mode and number of proofs:**
|    ``transpose_matrix(+list(list(number)),-list(list(number)))`` - ``one``


------------

.. index:: vector_value/3
.. _linear_algebra_protocol/0::vector_value/3:

``vector_value/3``
^^^^^^^^^^^^^^^^^^

Looks up a numeric vector element using a one-based index.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``vector_value(Vector,Index,Value)``
| **Mode and number of proofs:**
|    ``vector_value(+list(number),+integer,-number)`` - ``one``

| **Exceptions:**
|    ``Index`` is a variable:
|        ``instantiation_error``
|    ``Index`` is neither a variable nor an integer:
|        ``type_error(integer,Index)``
|    ``Index`` is an integer that is less than one:
|        ``domain_error(positive_integer,Index)``


------------

.. index:: matrix_row/3
.. _linear_algebra_protocol/0::matrix_row/3:

``matrix_row/3``
^^^^^^^^^^^^^^^^

Looks up a numeric matrix row using a one-based row index.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_row(Matrix,RowIndex,Row)``
| **Mode and number of proofs:**
|    ``matrix_row(+list(list(number)),+integer,-list(number))`` - ``one``

| **Exceptions:**
|    ``RowIndex`` is a variable:
|        ``instantiation_error``
|    ``RowIndex`` is neither a variable nor an integer:
|        ``type_error(integer,RowIndex)``
|    ``RowIndex`` is an integer that is less than one:
|        ``domain_error(positive_integer,RowIndex)``


------------

.. index:: matrix_column/3
.. _linear_algebra_protocol/0::matrix_column/3:

``matrix_column/3``
^^^^^^^^^^^^^^^^^^^

Looks up a numeric matrix column using a one-based column index.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_column(Matrix,ColumnIndex,Column)``
| **Mode and number of proofs:**
|    ``matrix_column(+list(list(number)),+integer,-list(number))`` - ``one``

| **Exceptions:**
|    ``ColumnIndex`` is a variable:
|        ``instantiation_error``
|    ``ColumnIndex`` is neither a variable nor an integer:
|        ``type_error(integer,ColumnIndex)``
|    ``ColumnIndex`` is an integer that is less than one:
|        ``domain_error(positive_integer,ColumnIndex)``


------------

.. index:: matrix_value/4
.. _linear_algebra_protocol/0::matrix_value/4:

``matrix_value/4``
^^^^^^^^^^^^^^^^^^

Looks up a numeric matrix element using one-based row and column indices.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_value(Matrix,RowIndex,ColumnIndex,Value)``
| **Mode and number of proofs:**
|    ``matrix_value(+list(list(number)),+integer,+integer,-number)`` - ``one``

| **Exceptions:**
|    ``RowIndex`` is a variable:
|        ``instantiation_error``
|    ``ColumnIndex`` is a variable:
|        ``instantiation_error``
|    ``RowIndex`` is neither a variable nor an integer:
|        ``type_error(integer,RowIndex)``
|    ``ColumnIndex`` is neither a variable nor an integer:
|        ``type_error(integer,ColumnIndex)``
|    ``RowIndex`` is an integer that is less than one:
|        ``domain_error(positive_integer,RowIndex)``
|    ``ColumnIndex`` is an integer that is less than one:
|        ``domain_error(positive_integer,ColumnIndex)``


------------

.. index:: matrix_diagonal/2
.. _linear_algebra_protocol/0::matrix_diagonal/2:

``matrix_diagonal/2``
^^^^^^^^^^^^^^^^^^^^^

Extracts the main diagonal of a numeric matrix as a vector.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_diagonal(Matrix,Diagonal)``
| **Mode and number of proofs:**
|    ``matrix_diagonal(+list(list(number)),-list(number))`` - ``one``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: matrix_diagonal/3
.. _linear_algebra_protocol/0::matrix_diagonal/3:

``matrix_diagonal/3``
^^^^^^^^^^^^^^^^^^^^^

Extracts the diagonal of a numeric matrix at the given offset, where positive offsets select superdiagonals and negative offsets select subdiagonals.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_diagonal(Matrix,Offset,Diagonal)``
| **Mode and number of proofs:**
|    ``matrix_diagonal(+list(list(number)),+integer,-list(number))`` - ``one_or_error``

| **Exceptions:**
|    ``Offset`` is a variable:
|        ``instantiation_error``
|    ``Offset`` is neither a variable nor an integer:
|        ``type_error(integer,Offset)``
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: diagonal_matrix/2
.. _linear_algebra_protocol/0::diagonal_matrix/2:

``diagonal_matrix/2``
^^^^^^^^^^^^^^^^^^^^^

Constructs a square diagonal matrix from a numeric vector of diagonal entries.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``diagonal_matrix(Diagonal,Matrix)``
| **Mode and number of proofs:**
|    ``diagonal_matrix(+list(number),-list(list(number)))`` - ``one``


------------

.. index:: diagonal_matrix/3
.. _linear_algebra_protocol/0::diagonal_matrix/3:

``diagonal_matrix/3``
^^^^^^^^^^^^^^^^^^^^^

Constructs the minimal square numeric matrix whose diagonal at the given offset matches the input vector.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``diagonal_matrix(Diagonal,Offset,Matrix)``
| **Mode and number of proofs:**
|    ``diagonal_matrix(+list(number),+integer,-list(list(number)))`` - ``one_or_error``

| **Exceptions:**
|    ``Offset`` is a variable:
|        ``instantiation_error``
|    ``Offset`` is neither a variable nor an integer:
|        ``type_error(integer,Offset)``


------------

.. index:: diagonal_matrix/4
.. _linear_algebra_protocol/0::diagonal_matrix/4:

``diagonal_matrix/4``
^^^^^^^^^^^^^^^^^^^^^

Constructs a square numeric matrix of the requested size whose diagonal at the given offset matches the input vector.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``diagonal_matrix(Diagonal,Offset,Size,Matrix)``
| **Mode and number of proofs:**
|    ``diagonal_matrix(+list(number),+integer,+integer,-list(list(number)))`` - ``one_or_error``

| **Exceptions:**
|    ``Offset`` is a variable:
|        ``instantiation_error``
|    ``Offset`` is neither a variable nor an integer:
|        ``type_error(integer,Offset)``
|    ``Size`` is a variable:
|        ``instantiation_error``
|    ``Size`` is neither a variable nor an integer:
|        ``type_error(integer,Size)``
|    ``Size`` is an integer that is less than zero:
|        ``domain_error(non_negative_integer,Size)``
|    ``Size`` is smaller than the minimum size required by ``Diagonal`` and ``Offset``:
|        ``domain_error(minimum_matrix_size(A),Size)``


------------

.. index:: matrix_trace/2
.. _linear_algebra_protocol/0::matrix_trace/2:

``matrix_trace/2``
^^^^^^^^^^^^^^^^^^

Computes the trace of a numeric matrix by summing its main diagonal entries.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_trace(Matrix,Trace)``
| **Mode and number of proofs:**
|    ``matrix_trace(+list(list(number)),-number)`` - ``one``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: shift_matrix_diagonal/3
.. _linear_algebra_protocol/0::shift_matrix_diagonal/3:

``shift_matrix_diagonal/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Shifts the main diagonal of a numeric matrix by the given value.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``shift_matrix_diagonal(Matrix,Shift,ShiftedMatrix)``
| **Mode and number of proofs:**
|    ``shift_matrix_diagonal(+list(list(number)),+number,-list(list(number)))`` - ``one``

| **Exceptions:**
|    ``Shift`` is a variable:
|        ``instantiation_error``
|    ``Shift`` is neither a variable nor a number:
|        ``type_error(number,Shift)``


------------

.. index:: upper_triangular_part/2
.. _linear_algebra_protocol/0::upper_triangular_part/2:

``upper_triangular_part/2``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Returns the upper-triangular part of a numeric matrix including the main diagonal.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``upper_triangular_part(Matrix,UpperTriangular)``
| **Mode and number of proofs:**
|    ``upper_triangular_part(+list(list(number)),-list(list(number)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: upper_triangular_part/3
.. _linear_algebra_protocol/0::upper_triangular_part/3:

``upper_triangular_part/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Returns the upper-triangular part of a numeric matrix using the given diagonal offset, where positive offsets exclude leading diagonals and negative offsets include additional subdiagonals.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``upper_triangular_part(Matrix,Offset,UpperTriangular)``
| **Mode and number of proofs:**
|    ``upper_triangular_part(+list(list(number)),+integer,-list(list(number)))`` - ``one_or_error``

| **Exceptions:**
|    ``Offset`` is a variable:
|        ``instantiation_error``
|    ``Offset`` is neither a variable nor an integer:
|        ``type_error(integer,Offset)``
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: lower_triangular_part/2
.. _linear_algebra_protocol/0::lower_triangular_part/2:

``lower_triangular_part/2``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Returns the lower-triangular part of a numeric matrix including the main diagonal.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``lower_triangular_part(Matrix,LowerTriangular)``
| **Mode and number of proofs:**
|    ``lower_triangular_part(+list(list(number)),-list(list(number)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: lower_triangular_part/3
.. _linear_algebra_protocol/0::lower_triangular_part/3:

``lower_triangular_part/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Returns the lower-triangular part of a numeric matrix using the given diagonal offset, where positive offsets include additional superdiagonals and negative offsets exclude leading diagonals below the main diagonal.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``lower_triangular_part(Matrix,Offset,LowerTriangular)``
| **Mode and number of proofs:**
|    ``lower_triangular_part(+list(list(number)),+integer,-list(list(number)))`` - ``one_or_error``

| **Exceptions:**
|    ``Offset`` is a variable:
|        ``instantiation_error``
|    ``Offset`` is neither a variable nor an integer:
|        ``type_error(integer,Offset)``
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: solve_linear_system/3
.. _linear_algebra_protocol/0::solve_linear_system/3:

``solve_linear_system/3``
^^^^^^^^^^^^^^^^^^^^^^^^^

Solves a non-singular square numeric linear system with a vector right-hand side using pivoted elimination.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``solve_linear_system(Matrix,Values,Solution)``
| **Mode and number of proofs:**
|    ``solve_linear_system(+list(list(number)),+list(number),-list(float))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a square matrix:
|        ``domain_error(square_matrix,Matrix)``
|    ``Values`` does not have the same length as the size of ``Matrix``:
|        ``domain_error(vector_length(A),Values)``
|    ``Matrix`` is singular:
|        ``evaluation_error(zero_divisor)``


------------

.. index:: solve_linear_systems/3
.. _linear_algebra_protocol/0::solve_linear_systems/3:

``solve_linear_systems/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^

Solves a non-singular square numeric linear system with a matrix right-hand side using pivoted elimination.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``solve_linear_systems(Matrix,RightHandSides,Solutions)``
| **Mode and number of proofs:**
|    ``solve_linear_systems(+list(list(number)),+list(list(number)),-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a square matrix:
|        ``domain_error(square_matrix,Matrix)``
|    ``RightHandSides`` does not have the same number of rows as the size of ``Matrix``:
|        ``domain_error(matrix_row_count(A),RightHandSides)``
|    ``Matrix`` is singular:
|        ``evaluation_error(zero_divisor)``


------------

.. index:: determinant/2
.. _linear_algebra_protocol/0::determinant/2:

``determinant/2``
^^^^^^^^^^^^^^^^^

Computes the determinant of a square numeric matrix.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``determinant(Matrix,Determinant)``
| **Mode and number of proofs:**
|    ``determinant(+list(list(number)),-number)`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a square matrix:
|        ``domain_error(square_matrix,Matrix)``


------------

.. index:: inverse_matrix/2
.. _linear_algebra_protocol/0::inverse_matrix/2:

``inverse_matrix/2``
^^^^^^^^^^^^^^^^^^^^

Computes the inverse of a non-singular square numeric matrix by solving one linear system per identity column.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``inverse_matrix(Matrix,Inverse)``
| **Mode and number of proofs:**
|    ``inverse_matrix(+list(list(number)),-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a square matrix:
|        ``domain_error(square_matrix,Matrix)``
|    ``Matrix`` is singular:
|        ``evaluation_error(zero_divisor)``


------------

.. index:: qr_decomposition/3
.. _linear_algebra_protocol/0::qr_decomposition/3:

``qr_decomposition/3``
^^^^^^^^^^^^^^^^^^^^^^

Computes a thin QR decomposition of a rectangular numeric matrix, returning orthogonal columns and an upper-trapezoidal factor.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``qr_decomposition(Matrix,Orthogonal,UpperTriangular)``
| **Mode and number of proofs:**
|    ``qr_decomposition(+list(list(number)),-list(list(float)),-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: least_squares/3
.. _linear_algebra_protocol/0::least_squares/3:

``least_squares/3``
^^^^^^^^^^^^^^^^^^^

Computes a least-squares solution for a rectangular numeric linear system using a pivoted orthogonal solve without forming normal equations.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``least_squares(Matrix,Values,Solution)``
| **Mode and number of proofs:**
|    ``least_squares(+list(list(number)),+list(number),-list(float))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``
|    ``Values`` does not have the same length as the number of rows of ``Matrix``:
|        ``domain_error(vector_length(A),Values)``


------------

.. index:: matrix_rank/2
.. _linear_algebra_protocol/0::matrix_rank/2:

``matrix_rank/2``
^^^^^^^^^^^^^^^^^

Estimates the rank of a numeric matrix using the default library numerical tolerance of 1.0e-12.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_rank(Matrix,Rank)``
| **Mode and number of proofs:**
|    ``matrix_rank(+list(list(number)),-integer)`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: matrix_rank/3
.. _linear_algebra_protocol/0::matrix_rank/3:

``matrix_rank/3``
^^^^^^^^^^^^^^^^^

Estimates the rank of a numeric matrix using the given non-negative numerical tolerance.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``matrix_rank(Matrix,Tolerance,Rank)``
| **Mode and number of proofs:**
|    ``matrix_rank(+list(list(number)),+number,-integer)`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: symmetric_eigen/3
.. _linear_algebra_protocol/0::symmetric_eigen/3:

``symmetric_eigen/3``
^^^^^^^^^^^^^^^^^^^^^

Computes an orthonormal eigendecomposition of a real symmetric dense matrix using the default library numerical tolerance of 1.0e-12.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``symmetric_eigen(Matrix,Eigenvectors,Eigenvalues)``
| **Mode and number of proofs:**
|    ``symmetric_eigen(+list(list(number)),-list(list(float)),-list(float))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a square matrix:
|        ``domain_error(square_matrix,Matrix)``
|    ``Matrix`` is not symmetric within the default numerical tolerance:
|        ``domain_error(symmetric_matrix,Matrix)``


------------

.. index:: symmetric_eigen/4
.. _linear_algebra_protocol/0::symmetric_eigen/4:

``symmetric_eigen/4``
^^^^^^^^^^^^^^^^^^^^^

Computes an orthonormal eigendecomposition of a real symmetric dense matrix using the given non-negative numerical tolerance.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``symmetric_eigen(Matrix,Tolerance,Eigenvectors,Eigenvalues)``
| **Mode and number of proofs:**
|    ``symmetric_eigen(+list(list(number)),+number,-list(list(float)),-list(float))`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``
|    ``Matrix`` is not a square matrix:
|        ``domain_error(square_matrix,Matrix)``
|    ``Matrix`` is not symmetric within ``Tolerance``:
|        ``domain_error(symmetric_matrix,Matrix)``


------------

.. index:: symmetric_eigen/5
.. _linear_algebra_protocol/0::symmetric_eigen/5:

``symmetric_eigen/5``
^^^^^^^^^^^^^^^^^^^^^

Computes an orthonormal eigendecomposition of a real symmetric dense matrix using the given non-negative numerical tolerance and positive iteration bound.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``symmetric_eigen(Matrix,Tolerance,MaximumIterations,Eigenvectors,Eigenvalues)``
| **Mode and number of proofs:**
|    ``symmetric_eigen(+list(list(number)),+number,+integer,-list(list(float)),-list(float))`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``
|    ``MaximumIterations`` is a variable:
|        ``instantiation_error``
|    ``MaximumIterations`` is neither a variable nor an integer:
|        ``type_error(integer,MaximumIterations)``
|    ``MaximumIterations`` is an integer that is less than one:
|        ``domain_error(positive_integer,MaximumIterations)``
|    ``Matrix`` is not a square matrix:
|        ``domain_error(square_matrix,Matrix)``
|    ``Matrix`` is not symmetric within ``Tolerance``:
|        ``domain_error(symmetric_matrix,Matrix)``


------------

.. index:: pseudo_inverse/2
.. _linear_algebra_protocol/0::pseudo_inverse/2:

``pseudo_inverse/2``
^^^^^^^^^^^^^^^^^^^^

Computes the Moore-Penrose pseudo-inverse of a rectangular numeric matrix using the default library numerical tolerance of 1.0e-12.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``pseudo_inverse(Matrix,PseudoInverse)``
| **Mode and number of proofs:**
|    ``pseudo_inverse(+list(list(number)),-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: pseudo_inverse/3
.. _linear_algebra_protocol/0::pseudo_inverse/3:

``pseudo_inverse/3``
^^^^^^^^^^^^^^^^^^^^

Computes the Moore-Penrose pseudo-inverse of a rectangular numeric matrix using the given non-negative numerical tolerance.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``pseudo_inverse(Matrix,Tolerance,PseudoInverse)``
| **Mode and number of proofs:**
|    ``pseudo_inverse(+list(list(number)),+number,-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: null_space/2
.. _linear_algebra_protocol/0::null_space/2:

``null_space/2``
^^^^^^^^^^^^^^^^

Computes an orthonormal basis for the right null space of a rectangular numeric matrix using the default library numerical tolerance of 1.0e-12.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``null_space(Matrix,Basis)``
| **Mode and number of proofs:**
|    ``null_space(+list(list(number)),-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: null_space/3
.. _linear_algebra_protocol/0::null_space/3:

``null_space/3``
^^^^^^^^^^^^^^^^

Computes an orthonormal basis for the right null space of a rectangular numeric matrix using the given non-negative numerical tolerance.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``null_space(Matrix,Tolerance,Basis)``
| **Mode and number of proofs:**
|    ``null_space(+list(list(number)),+number,-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``
|    ``Matrix`` is not a rectangular matrix:
|        ``domain_error(rectangular_matrix,Matrix)``


------------

.. index:: normalize_vector/2
.. _linear_algebra_protocol/0::normalize_vector/2:

``normalize_vector/2``
^^^^^^^^^^^^^^^^^^^^^^

Normalizes a numeric vector to unit length when its norm is above the default library numerical tolerance of 1.0e-12.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``normalize_vector(Vector,NormalizedVector)``
| **Mode and number of proofs:**
|    ``normalize_vector(+list(number),-list(number))`` - ``one``


------------

.. index:: normalize_vector/3
.. _linear_algebra_protocol/0::normalize_vector/3:

``normalize_vector/3``
^^^^^^^^^^^^^^^^^^^^^^

Normalizes a numeric vector to unit length when its norm is above the given non-negative numerical tolerance.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``normalize_vector(Vector,Tolerance,NormalizedVector)``
| **Mode and number of proofs:**
|    ``normalize_vector(+list(number),+number,-list(number))`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``


------------

.. index:: difference_norm/3
.. _linear_algebra_protocol/0::difference_norm/3:

``difference_norm/3``
^^^^^^^^^^^^^^^^^^^^^

Computes the Euclidean norm of the difference between two numeric vectors.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``difference_norm(Vector1,Vector2,Norm)``
| **Mode and number of proofs:**
|    ``difference_norm(+list(number),+list(number),-float)`` - ``one``


------------

.. index:: stabilize_vector_sign/2
.. _linear_algebra_protocol/0::stabilize_vector_sign/2:

``stabilize_vector_sign/2``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Normalizes a vector sign convention by flipping vectors whose first significant component is negative.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``stabilize_vector_sign(Vector,StableVector)``
| **Mode and number of proofs:**
|    ``stabilize_vector_sign(+list(number),-list(number))`` - ``one``


------------

.. index:: stabilize_vector_sign/3
.. _linear_algebra_protocol/0::stabilize_vector_sign/3:

``stabilize_vector_sign/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Normalizes a vector sign convention by flipping vectors whose first significant component under the given non-negative numerical tolerance is negative.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``stabilize_vector_sign(Vector,Tolerance,StableVector)``
| **Mode and number of proofs:**
|    ``stabilize_vector_sign(+list(number),+number,-list(number))`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``


------------

.. index:: first_significant_component/2
.. _linear_algebra_protocol/0::first_significant_component/2:

``first_significant_component/2``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Returns the first component whose absolute value exceeds the default library numerical tolerance of 1.0e-12, defaulting to zero when no such component exists.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``first_significant_component(Vector,First)``
| **Mode and number of proofs:**
|    ``first_significant_component(+list(number),-number)`` - ``one``


------------

.. index:: first_significant_component/3
.. _linear_algebra_protocol/0::first_significant_component/3:

``first_significant_component/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Returns the first component whose absolute value exceeds the given non-negative numerical tolerance, defaulting to zero when no such component exists.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``first_significant_component(Vector,Tolerance,First)``
| **Mode and number of proofs:**
|    ``first_significant_component(+list(number),+number,-number)`` - ``one_or_error``

| **Exceptions:**
|    ``Tolerance`` is a variable:
|        ``instantiation_error``
|    ``Tolerance`` is neither a variable nor a number:
|        ``type_error(number,Tolerance)``
|    ``Tolerance`` is a number but not a non-negative number:
|        ``domain_error(non_negative_number,Tolerance)``


------------

.. index:: add_scaled_outer_product/4
.. _linear_algebra_protocol/0::add_scaled_outer_product/4:

``add_scaled_outer_product/4``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Adds a scaled outer product of a vector with itself to an accumulated matrix.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``add_scaled_outer_product(Vector,Scale,Matrix0,Matrix)``
| **Mode and number of proofs:**
|    ``add_scaled_outer_product(+list(number),+number,+list(list(number)),-list(list(number)))`` - ``one``

| **Exceptions:**
|    ``Scale`` is a variable:
|        ``instantiation_error``
|    ``Scale`` is neither a variable nor a number:
|        ``type_error(number,Scale)``


------------

.. index:: covariance_matrix/2
.. _linear_algebra_protocol/0::covariance_matrix/2:

``covariance_matrix/2``
^^^^^^^^^^^^^^^^^^^^^^^

Computes the sample covariance matrix from a list of centered numeric row vectors.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``covariance_matrix(Rows,CovarianceMatrix)``
| **Mode and number of proofs:**
|    ``covariance_matrix(+list(list(number)),-list(list(number)))`` - ``one_or_error``

| **Exceptions:**
|    ``Rows`` contains fewer than two rows:
|        ``domain_error(minimum_number_of_rows(2),Rows)``


------------

.. index:: cholesky_decomposition/2
.. _linear_algebra_protocol/0::cholesky_decomposition/2:

``cholesky_decomposition/2``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Computes the lower-triangular Cholesky factor of a positive definite numeric matrix.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``cholesky_decomposition(Matrix,CholeskyFactor)``
| **Mode and number of proofs:**
|    ``cholesky_decomposition(+list(list(number)),-list(list(float)))`` - ``one_or_error``

| **Exceptions:**
|    ``Matrix`` is not numerically positive definite:
|        ``domain_error(positive_definite_matrix,Matrix)``


------------

.. index:: solve_cholesky/3
.. _linear_algebra_protocol/0::solve_cholesky/3:

``solve_cholesky/3``
^^^^^^^^^^^^^^^^^^^^

Solves a positive definite linear system from a lower-triangular Cholesky factor and a right-hand-side vector.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``solve_cholesky(CholeskyFactor,Values,Solution)``
| **Mode and number of proofs:**
|    ``solve_cholesky(+list(list(number)),+list(number),-list(float))`` - ``one``


------------

.. index:: forward_substitution/3
.. _linear_algebra_protocol/0::forward_substitution/3:

``forward_substitution/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^

Solves a lower-triangular system by forward substitution.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``forward_substitution(LowerTriangular,Values,Solution)``
| **Mode and number of proofs:**
|    ``forward_substitution(+list(list(number)),+list(number),-list(float))`` - ``one``


------------

.. index:: solve_lower_triangular_matrix/3
.. _linear_algebra_protocol/0::solve_lower_triangular_matrix/3:

``solve_lower_triangular_matrix/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Solves a lower-triangular system with a matrix right-hand side by forward substitution on each column.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``solve_lower_triangular_matrix(LowerTriangular,Matrix,Solution)``
| **Mode and number of proofs:**
|    ``solve_lower_triangular_matrix(+list(list(number)),+list(list(number)),-list(list(float)))`` - ``one``


------------

.. index:: solve_upper_triangular/3
.. _linear_algebra_protocol/0::solve_upper_triangular/3:

``solve_upper_triangular/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Solves an upper-triangular system by backward substitution.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``solve_upper_triangular(UpperTriangular,Values,Solution)``
| **Mode and number of proofs:**
|    ``solve_upper_triangular(+list(list(number)),+list(number),-list(float))`` - ``one``


------------

.. index:: solve_upper_triangular_matrix/3
.. _linear_algebra_protocol/0::solve_upper_triangular_matrix/3:

``solve_upper_triangular_matrix/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Solves an upper-triangular system with a matrix right-hand side by backward substitution on each column.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``solve_upper_triangular_matrix(UpperTriangular,Matrix,Solution)``
| **Mode and number of proofs:**
|    ``solve_upper_triangular_matrix(+list(list(number)),+list(list(number)),-list(list(float)))`` - ``one``


------------

.. index:: backward_substitution/3
.. _linear_algebra_protocol/0::backward_substitution/3:

``backward_substitution/3``
^^^^^^^^^^^^^^^^^^^^^^^^^^^

Solves the transposed upper-triangular system induced by a lower-triangular Cholesky factor using backward substitution.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``backward_substitution(CholeskyFactor,Values,Solution)``
| **Mode and number of proofs:**
|    ``backward_substitution(+list(list(number)),+list(number),-list(float))`` - ``one``


------------

.. index:: invert_from_cholesky/2
.. _linear_algebra_protocol/0::invert_from_cholesky/2:

``invert_from_cholesky/2``
^^^^^^^^^^^^^^^^^^^^^^^^^^

Computes a matrix inverse from a Cholesky factor by solving one linear system per basis column.

| **Compilation flags:**
|    ``static``

| **Template:**
|    ``invert_from_cholesky(CholeskyFactor,Inverse)``
| **Mode and number of proofs:**
|    ``invert_from_cholesky(+list(list(number)),-list(list(float)))`` - ``one``


------------

Protected predicates
--------------------

(none)

Private predicates
------------------

(none)

Operators
---------

(none)