general_linear_solver.py

def paqlu_decomposition_in_place(A):
    """
    Compute an in-place PAQ = LU decomposition of a rectangular matrix A with
    row and column pivoting. This is a stub: the function body is not
    implemented here, but the docstring fully specifies the interface,
    conventions, and expected behavior so an implementation can be written.

    Purpose and convention
    - For a given input matrix A (m-by-n), produce integer permutation vectors
      P (length m) and Q (length n) and overwrite A in place so that
      if A_original denotes the input before modification, then
          P_matrix @ A_original @ Q_matrix = L @ U,
      where P_matrix and Q_matrix are the permutation matrices associated
      with the vectors P and Q (see "Permutation vectors" below), L is an
      m-by-r lower trapezoidal matrix with unit diagonal in its leading r
      rows (where r = numerical rank), and U is an r-by-n upper trapezoidal
      matrix. The factors L and U are stored in the single array A as
      described in "Storage convention" below.

    Parameters
    ----------
    A : numpy.ndarray, shape (m, n)
        The input matrix to be decomposed. It will be modified in place to
        contain the L and U factors. A must be a two-dimensional contiguous
        numeric array (float or complex). The function does not allocate a
        separate output copy for the factorization results.

    Returns
    -------
    P : numpy.ndarray, shape (m,), dtype=int
        Row-permutation vector. The permutation vector uses zero-based
        indexing and satisfies
            A_original[P, :] == P_matrix @ A_original
        where P_matrix is the corresponding m-by-m permutation matrix.
        In words, row i of the permuted matrix is row P[i] of the original.

    Q : numpy.ndarray, shape (n,), dtype=int
        Column-permutation vector. The permutation vector uses zero-based
        indexing and satisfies
            A_original[:, Q] == A_original @ Q_matrix
        where Q_matrix is the corresponding n-by-n permutation matrix.
        In words, column j of the permuted matrix is column Q[j] of the original.

    A : numpy.ndarray, shape (m, n)
        The same array object that was passed in, modified in place so that
        it contains the L and U factors:
          - U is stored in the upper triangle (including the diagonal) of A
            in the first r rows used by the elimination.
          - The strict lower triangle of A contains the multipliers (entries
            of L below the unit diagonal). The unit diagonal of L is implicit
            and not stored (i.e., L[i,i] = 1 for i < r).
        The exact layout for rectangular shapes:
          - If m >= n (tall or square): U is n-by-n stored in rows 0..n-1,
            L is m-by-n with unit diagonal in its first n rows.
          - If m < n (wide): U is m-by-n stored in rows 0..m-1,
            L is m-by-m with unit diagonal in its first m rows.
        The numerical rank r is determined during elimination by pivot
        thresholding (see "Rank and tolerance" below).

    Notes on algorithm
    - The decomposition is computed by Gaussian elimination with:
        * row pivoting to avoid division by small numbers (partial pivoting),
        * column pivoting simulated via permutation vector Q so that pivot
          columns are moved before non-pivot columns (this is useful for
          identifying pivot columns and free variables).
    - Column pivoting should be implemented so that pivot columns are
      selected with a criterion analogous to partial pivoting on rows:
      choose a column that contains a sufficiently large pivot in the
      current working submatrix. This places "pivot columns" before
      "non-pivot (free) columns" in the order given by Q.
    - The routine must not allocate full dense permutation matrices P_matrix
      or Q_matrix; only the integer permutation vectors P and Q are returned.

    Rank and tolerance
    - The algorithm must decide when a pivot is numerically zero. An
      appropriate default is to compare |pivot| to tol = max(m,n) * eps * max_abs,
      where eps is machine precision and max_abs is the maximum absolute
      value in the current working submatrix. The implementation may accept a
      user-specified tolerance parameter; if not, it must document the
      default used.
    - The integer r (numerical rank) is the number of successful pivots
      performed; after r pivots the remaining columns are considered free.

    Complexity
    - Time: O(min(m,n) * m * n) in the dense case (standard Gaussian
      elimination complexity with pivoting).
    - Memory: O(1) extra beyond the input A and the two permutation vectors.

    Stability and usage
    - This routine is intended for exact-solve and nullspace computations on
      moderately sized dense matrices. For very large matrices or ill-conditioned
      problems, use a robust SVD-based method to compute nullspaces and
      least-squares solutions.
    - After calling this function, use the returned P, Q, and in-place A
      to form solutions, compute nullspace basis vectors, or to apply
      forward/back substitution.

    Exceptions
    - The function should raise a ValueError if A is not two-dimensional.
    - The function should raise TypeError if A's dtype is not a supported
      numeric type.

    Example (conceptual)
    - Suppose A0 is the original m-by-n array. After calling
          P, Q, A = paqlu_decomposition_in_place(A0)
      the client can interpret the factorization as:
          (permute rows by P) and (permute columns by Q) to get L and U
      i.e. A0[P, :][:, Q] == L @ U (modulo rounding).
    """
    raise NotImplementedError("paqlu_decomposition_in_place is a stub; implement factorization here.")


def solve(A, b):
    """
    Compute a parametric solution of the linear system A x = b in the form
        x = N @ xfree + c,
    where xfree is an arbitrary vector of free-variable parameters. This is
    a stub: the function body is not implemented here, but the docstring
    precisely specifies the inputs, outputs, conventions, and error behavior.

    Given
    - A: an m-by-n matrix,
    - b: a length-m vector (or m-by-k matrix for multiple right-hand sides),

    this function returns
    - N: an n-by(f) matrix whose columns form a basis for the nullspace of A,
         i.e., A @ N == 0 (up to numerical tolerance). Here f = n - r is
         the number of free variables and r is the numerical rank of A.
    - c: a length-n vector (or n-by-k matrix matching b's second dimension)
         that is a particular solution of A x = b, i.e., A @ c == b (within
         tolerance), provided the system is consistent.

    Parameters
    ----------
    A : numpy.ndarray, shape (m, n)
        Coefficient matrix of the linear system. A may be modified in place
        by the routine that computes an LU-like factorization; if the user
        wishes to preserve A, they should pass a copy.

    b : numpy.ndarray, shape (m,) or (m, k)
        Right-hand side vector (or multiple right-hand sides). Entries must
        be numeric and compatible with the dtype of A.

    Returns
    -------
    N : numpy.ndarray, shape (n, f)
        Basis for the nullspace (homogeneous solutions) of A. If f == 0 (full
        column rank), N is an array with shape (n, 0). Columns of N are
        linearly independent and span {x : A @ x == 0}.

    c : numpy.ndarray, shape (n,) or (n, k)
        A particular solution of A x = b. If multiple right-hand sides are
        provided (b shape (m,k)), c has shape (n,k). If the system is
        inconsistent (no solution), the function should raise a ValueError.

    Algorithm outline (implementation guidance)
    1. Compute a PAQ = LU decomposition of A using paqlu_decomposition_in_place
       (or an equivalent routine) to identify pivot columns and the numerical
       rank r. The function paqlu_decomposition_in_place should return the row
       and column permutation vectors P, Q and modify A in place to store L and U.
    2. Apply the same row permutations to b: b_permuted = b[P].
    3. Solve L y = b_permuted by forward substitution for the first r rows.
    4. Solve U z = y_prefix by back substitution for the pivot variables.
       If the system is inconsistent (e.g., a zero row in U corresponds to a
       nonzero entry in y), raise ValueError("inconsistent system").
    5. Construct a particular solution in the permuted variable ordering:
          x_perm = [z (pivot variables); 0 (free variables)]
       Because column permutation Q was used, place pivot and free variables
       into their original positions by applying the inverse permutation:
          c = inverse_permute_columns(x_perm, Q)
    6. Build a nullspace basis N by setting each free variable to 1 (one at a
       time) and solving the triangular system for pivot variables (similar to
       computing the reduced-column-echelon homogeneous solutions). Then map
       back through inverse column permutation Q so that each basis vector is
       expressed in the original variable ordering.

    Numerical tolerances
    - All comparisons to zero should use a tolerance based on machine
      precision, matrix norms, and the scale of the problem (see docstring
      of paqlu_decomposition_in_place for a recommended default).

    Edge cases and exceptions
    - If A has shape (0, n) or (m, 0) handle appropriately:
        * m == 0: any b must be empty; then any x is a solution -> choose c = 0,
          N = identity (n-by-n).
        * n == 0: only possible if b == 0; otherwise inconsistent.
    - If the system is inconsistent (no x satisfies A x = b within tolerance),
      raise ValueError("inconsistent system: A x = b has no solution").
    - If b has multiple right-hand sides, compute corresponding columns of c
      and return an N that is common for all right-hand sides.

    Complexity
    - Dominated by the cost of the decomposition: O(min(m,n) * m * n) time,
      plus O(m*n) for forward/back substitution steps.

    Examples (conceptual)
    - For a full-column-rank tall matrix (m >= n, rank = n), f = 0 and
      N has shape (n, 0). The routine computes the unique solution c = A^{-1} b
      via the LU factors.
    - For an underdetermined system (n > m, rank = r < n), f = n - r > 0,
      and the returned N provides a basis for the affine family of solutions.

    Returns
    -------
    N, c : numpy.ndarray, numpy.ndarray

    Notes
    - This stub assumes an implementation will rely on paqlu_decomposition_in_place.
    - The function should preserve shapes and dtypes consistently; if b is
      real and A is real, return real N and c; if complex, return complex types.

    """
    raise NotImplementedError("solve is a stub; implement parametric solver here.")