SciPy 0.6.0 API Documentation Generated by Endo, 2007-10-17

Variables

• eps

  eps = numpy.finfo(float).eps
  
  

• feps

  feps = numpy.finfo(single).eps
  
  

Function summary

• all_mat(*args)
• cho_solve((c, lower), b, overwrite_b = 0)
• cholesky_banded(ab, overwrite_ab = 0, lower = 0)
• det(a, overwrite_a = 0)
• hankel(c, r = None)
• inv(a, overwrite_a = 0)
• kron(a, b)
• lstsq(a, b, cond = None, overwrite_a = 0, overwrite_b = 0)
• lu_solve((lu, piv), b, trans = 0, overwrite_b = 0)
• norm(x, ord = None)
• pinv(a, cond = None, rcond = None)
• pinv2(a, cond = None, rcond = None)
• solve(a, b, sym_pos = 0, lower = 0, overwrite_a = 0, overwrite_b = 0, debug = 0)
• solve_banded((l, u), ab, b, overwrite_ab = 0, overwrite_b = 0, debug = 0)
• solveh_banded(ab, b, overwrite_ab = 0, overwrite_b = 0, lower = 0)
• toeplitz(c, r = None)
• tri(N, M = None, k = 0, dtype = None)
• tril(m, k = 0)
• triu(m, k = 0)

Functions

• all_mat(*args)
• cho_solve((c, lower), b, overwrite_b = 0)

  cho_solve((c, lower), b, overwrite_b=0) -> x

  Solve a system of equations given a previously cholesky factored matrix

  Inputs:

  (c,lower) -- The factored matrix, a (the output of cho_factor) b -- a set of right-hand sides

  Outputs:

  x -- the solution to the system a*x = b

• cholesky_banded(ab, overwrite_ab = 0, lower = 0)

  cholesky_banded(ab, overwrite_ab=0, lower=0) -> c
  
  Compute the Cholesky decomposition of a 	
  banded symmetric or Hermitian positive definite
  matrix stored in lower diagonal ordered form (lower=1)
  
  a11 a22 a33 a44 a55 a66
  a21 a32 a43 a54 a65 *
  a31 a42 a53 a64 *   *
  
  or upper diagonal ordered form
  
  *   *   a31 a42 a53 a64
  *   a21 a32 a43 a54 a65
  a11 a22 a33 a44 a55 a66
  
  Inputs:
  
    ab -- An N x l
    overwrite_ab - Discard data in ab
    lower - is ab in lower or upper form?
  
  Outputs:  
  
    c:  the Cholesky factorization of ab
  
  

• det(a, overwrite_a = 0)

  det(a, overwrite_a=0) -> d

  Return determinant of a square matrix.

• hankel(c, r = None)

  Construct a hankel matrix (i.e. matrix with constant anti-diagonals).

  Description:

  hankel(c,r) is a Hankel matrix whose first column is c and whose last row is r.

  hankel(c) is a square Hankel matrix whose first column is C. Elements below the first anti-diagonal are zero.

  See also: toeplitz

• inv(a, overwrite_a = 0)

  inv(a, overwrite_a=0) -> a_inv

  Return inverse of square matrix a.

• kron(a, b)

  kronecker product of a and b
  
  Kronecker product of two matrices is block matrix
  [[ a[ 0 ,0]*b, a[ 0 ,1]*b, ... , a[ 0 ,n-1]*b  ],
   [ ...                                   ...   ],
   [ a[m-1,0]*b, a[m-1,1]*b, ... , a[m-1,n-1]*b  ]]
  

• lstsq(a, b, cond = None, overwrite_a = 0, overwrite_b = 0)

  lstsq(a, b, cond=None, overwrite_a=0, overwrite_b=0) -> x,resids,rank,s
  
  Return least-squares solution of a * x = b.
  
  Inputs:
  
    a -- An M x N matrix.
    b -- An M x nrhs matrix or M vector.
    cond -- Used to determine effective rank of a.
  
  Outputs:
  
    x -- The solution (N x nrhs matrix) to the minimization problem:
                2-norm(| b - a * x |) -> min
    resids -- The residual sum-of-squares for the solution matrix x
              (only if M>N and rank==N).
    rank -- The effective rank of a.
    s -- Singular values of a in decreasing order. The condition number
         of a is abs(s[0]/s[-1]).
  

• lu_solve((lu, piv), b, trans = 0, overwrite_b = 0)

  lu_solve((lu, piv), b, trans=0, overwrite_b=0) -> x
  
  Solve a system of equations given a previously factored matrix
  
  Inputs:
  
    (lu,piv) -- The factored matrix, a (the output of lu_factor)
    b        -- a set of right-hand sides
    trans    -- type of system to solve:
                0 : a   * x = b   (no transpose)
                1 : a^T * x = b   (transpose)
                2   a^H * x = b   (conjugate transpose)
  
  Outputs:
  
     x -- the solution to the system
  

• norm(x, ord = None)

  norm(x, ord=None) -> n

  Matrix or vector norm.

  Inputs:

  x -- a rank-1 (vector) or rank-2 (matrix) array ord -- the order of the norm.

  Comments:

  For arrays of any rank, if ord is None:

  calculate the square norm (Euclidean norm for vectors, Frobenius norm for matrices)

  For vectors ord can be any real number including Inf or -Inf.

  ord = Inf, computes the maximum of the magnitudes ord = -Inf, computes minimum of the magnitudes ord is finite, computes sum(abs(x)**ord,axis=0)**(1.0/ord)

  For matrices ord can only be one of the following values:

  ord = 2 computes the largest singular value ord = -2 computes the smallest singular value ord = 1 computes the largest column sum of absolute values ord = -1 computes the smallest column sum of absolute values ord = Inf computes the largest row sum of absolute values ord = -Inf computes the smallest row sum of absolute values ord = 'fro' computes the frobenius norm sqrt(sum(diag(X.H * X),axis=0))

  For values ord < 0, the result is, strictly speaking, not a mathematical 'norm', but it may still be useful for numerical purposes.

• pinv(a, cond = None, rcond = None)

  pinv(a, rcond=None) -> a_pinv

  Compute generalized inverse of A using least-squares solver.

• pinv2(a, cond = None, rcond = None)

  pinv2(a, rcond=None) -> a_pinv

  Compute the generalized inverse of A using svd.

• solve(a, b, sym_pos = 0, lower = 0, overwrite_a = 0, overwrite_b = 0, debug = 0)

  solve(a, b, sym_pos=0, lower=0, overwrite_a=0, overwrite_b=0) -> x
  
  Solve a linear system of equations a * x = b for x.
  
  Inputs:
  
    a -- An N x N matrix.
    b -- An N x nrhs matrix or N vector.
    sym_pos -- Assume a is symmetric and positive definite.
    lower -- Assume a is lower triangular, otherwise upper one.
             Only used if sym_pos is true.
    overwrite_y - Discard data in y, where y is a or b.
  
  Outputs:
  
    x -- The solution to the system a * x = b
  

• solve_banded((l, u), ab, b, overwrite_ab = 0, overwrite_b = 0, debug = 0)

  solve_banded((l,u), ab, b, overwrite_ab=0, overwrite_b=0) -> x
  
  Solve a linear system of equations a * x = b for x where
  a is a banded matrix stored in diagonal orded form
  
   *   *     a1u
  
   *  a12 a23 ...
  a11 a22 a33 ...
  a21 a32 a43 ...
  .
  al1 ..         *
  
  Inputs:
  
    (l,u) -- number of non-zero lower and upper diagonals, respectively.
    a -- An N x (l+u+1) matrix.
    b -- An N x nrhs matrix or N vector.
    overwrite_y - Discard data in y, where y is ab or b.
  
  Outputs:
  
    x -- The solution to the system a * x = b
  

• solveh_banded(ab, b, overwrite_ab = 0, overwrite_b = 0, lower = 0)

  solveh_banded(ab, b, overwrite_ab=0, overwrite_b=0) -> c, x
  
  Solve a linear system of equations a * x = b for x where
  a is a banded symmetric or Hermitian positive definite
  matrix stored in lower diagonal ordered form (lower=1)
  
  a11 a22 a33 a44 a55 a66
  a21 a32 a43 a54 a65 *
  a31 a42 a53 a64 *   *
  
  or upper diagonal ordered form
  
  *   *   a31 a42 a53 a64
  *   a21 a32 a43 a54 a65
  a11 a22 a33 a44 a55 a66
  
  Inputs:
  
    ab -- An N x l
    b -- An N x nrhs matrix or N vector.
    overwrite_y - Discard data in y, where y is ab or b.
    lower - is ab in lower or upper form?
  
  Outputs: 
  
    c:  the Cholesky factorization of ab
    x:  the solution to ab * x = b
  
  

• toeplitz(c, r = None)

  Construct a toeplitz matrix (i.e. a matrix with constant diagonals).

  Description:

  toeplitz(c,r) is a non-symmetric Toeplitz matrix with c as its first column and r as its first row.

  toeplitz(c) is a symmetric (Hermitian) Toeplitz matrix (r=c).

  See also: hankel

• tri(N, M = None, k = 0, dtype = None)

  returns a N-by-M matrix where all the diagonals starting from lower left corner up to the k-th are all ones.

• tril(m, k = 0)

  returns the elements on and below the k-th diagonal of m. k=0 is the main diagonal, k > 0 is above and k < 0 is below the main diagonal.

• triu(m, k = 0)

  returns the elements on and above the k-th diagonal of m. k=0 is the main diagonal, k > 0 is above and k < 0 is below the main diagonal.

Imported Names