matfuncs.py

#
# Author: Travis Oliphant, March 2002
#

__all__ = ['expm','expm2','expm3','cosm','sinm','tanm','coshm','sinhm',
           'tanhm','logm','funm','signm','sqrtm']

from numpy import asarray, Inf, dot, floor, eye, diag, exp, \
     product, logical_not, ravel, transpose, conjugate, \
     cast, log, ogrid, imag, real, absolute, amax, sign, \
     isfinite, sqrt, identity, single, ceil, log2
from numpy import matrix as mat
import numpy as np

from scipy.sparse.base import isspmatrix
from scipy.sparse.construct import eye as speye

# Local imports
from misc import norm
from basic import solve, inv
from special_matrices import triu, all_mat
from decomp import eig
from decomp_svd import orth, svd
from decomp_schur import schur, rsf2csf
import warnings

eps = np.finfo(float).eps
feps = np.finfo(single).eps

def expm(A, q=False):
    """Compute the matrix exponential using Pade approximation.

    Parameters
    ----------
    A : array, shape(M,M)
        Matrix to be exponentiated

    Returns
    -------
    expA : array, shape(M,M)
        Matrix exponential of A

    References
    ----------
    N. J. Higham,
    "The Scaling and Squaring Method for the Matrix Exponential Revisited",
    SIAM. J. Matrix Anal. & Appl. 26, 1179 (2005).

    """
    if q: warnings.warn("argument q=... in scipy.linalg.expm is deprecated.")
    A = asarray(A)
    A_L1 = norm(A,1)
    n_squarings = 0

    if A.dtype == 'float64' or A.dtype == 'complex128':
        if A_L1 < 1.495585217958292e-002:
            U,V = _pade3(A)
        elif A_L1 < 2.539398330063230e-001:
            U,V = _pade5(A)
        elif A_L1 < 9.504178996162932e-001:
            U,V = _pade7(A)
        elif A_L1 < 2.097847961257068e+000:
            U,V = _pade9(A)
        else:
            maxnorm = 5.371920351148152
            n_squarings = max(0, int(ceil(log2(A_L1 / maxnorm))))
            A = A / 2**n_squarings
            U,V = _pade13(A)
    elif A.dtype == 'float32' or A.dtype == 'complex64':
        if A_L1 < 4.258730016922831e-001:
            U,V = _pade3(A)
        elif A_L1 < 1.880152677804762e+000:
            U,V = _pade5(A)
        else:
            maxnorm = 3.925724783138660
            n_squarings = max(0, int(ceil(log2(A_L1 / maxnorm))))
            A = A / 2**n_squarings
            U,V = _pade7(A)
    else:
        raise ValueError("invalid type: "+str(A.dtype))

    P = U + V  # p_m(A) : numerator
    Q = -U + V # q_m(A) : denominator

    R = solve(Q,P)
    # squaring step to undo scaling
    for i in range(n_squarings):
        R = dot(R,R)

    return R

# implementation of Pade approximations of various degree using the algorithm presented in [Higham 2005]
# These should apply to both dense and sparse matricies.
def _pade3(A):
    b = (120., 60., 12., 1.)
    if isspmatrix(A):
        ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
    else:
        ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
    A2 = A.dot(A)
    U = A.dot(b[3]*A2 + b[1]*ident)
    V = b[2]*A2 + b[0]*ident
    return U,V

def _pade5(A):
    b = (30240., 15120., 3360., 420., 30., 1.)
    if isspmatrix(A):
        ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
    else:
        ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
    A2 = A.dot(A)
    A4 = A2.dot(A2)
    U = A.dot(b[5]*A4 + b[3]*A2 + b[1]*ident)
    V = b[4]*A4 + b[2]*A2 + b[0]*ident
    return U,V

def _pade5(A):
    b = (30240., 15120., 3360., 420., 30., 1.)
    if isspmatrix(A):
        ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
    else:
        ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
    A2 = A.dot(A)
    A4 = A2.dot(A2)
    U = A.dot(b[5]*A4 + b[3]*A2 + b[1]*ident)
    V = b[4]*A4 + b[2]*A2 + b[0]*ident
    return U,V

def _pade7(A):
    b = (17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.)
    if isspmatrix(A):
        ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
    else:
        ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
    A2 = A.dot(A)
    A4 = A2.dot(A2)
    A6 = A4.dot(A2)
    U = A.dot(b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident)
    V = b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident
    return U,V

def _pade9(A):
    b = (17643225600., 8821612800., 2075673600., 302702400., 30270240.,
                2162160., 110880., 3960., 90., 1.)
    if isspmatrix(A):
        ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
    else:
        ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
    A2 = A.dot(A)
    A4 = A2.dot(A2)
    A6 = A4.dot(A2)
    A8 = A6.dot(A2)
    U = A.dot(b[9]*A8 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident)
    V = b[8]*A8 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident
    return U,V

def _pade13(A):
    b = (64764752532480000., 32382376266240000., 7771770303897600.,
    1187353796428800., 129060195264000., 10559470521600., 670442572800.,
    33522128640., 1323241920., 40840800., 960960., 16380., 182., 1.)
    if isspmatrix(A):
        ident = speye(A.shape[0], A.shape[1], dtype=A.dtype, format=A.format)
    else:
        ident = eye(A.shape[0], A.shape[1], dtype=A.dtype)
    A2 = A.dot(A)
    A4 = A2.dot(A2)
    A6 = A4.dot(A2)
    U = A.dot(A6.dot(b[13]*A6 + b[11]*A4 + b[9]*A2) + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident)
    V = A6.dot(b[12]*A6 + b[10]*A4 + b[8]*A2) + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident
    return U,V

def expm2(A):
    """Compute the matrix exponential using eigenvalue decomposition.

    Parameters
    ----------
    A : array, shape(M,M)
        Matrix to be exponentiated

    Returns
    -------
    expA : array, shape(M,M)
        Matrix exponential of A

    """
    A = asarray(A)
    t = A.dtype.char
    if t not in ['f','F','d','D']:
        A = A.astype('d')
        t = 'd'
    s,vr = eig(A)
    vri = inv(vr)
    r = dot(dot(vr,diag(exp(s))),vri)
    if t in ['f', 'd']:
        return r.real.astype(t)
    else:
        return r.astype(t)

def expm3(A, q=20):
    """Compute the matrix exponential using Taylor series.

    Parameters
    ----------
    A : array, shape(M,M)
        Matrix to be exponentiated
    q : integer
        Order of the Taylor series

    Returns
    -------
    expA : array, shape(M,M)
        Matrix exponential of A

    """
    A = asarray(A)
    t = A.dtype.char
    if t not in ['f','F','d','D']:
        A = A.astype('d')
        t = 'd'
    A = mat(A)
    eA = eye(*A.shape,**{'dtype':t})
    trm = mat(eA, copy=True)
    castfunc = cast[t]
    for k in range(1,q):
        trm *= A / castfunc(k)
        eA += trm
    return eA

_array_precision = {'i': 1, 'l': 1, 'f': 0, 'd': 1, 'F': 0, 'D': 1}

def toreal(arr, tol=None):
    """Return as real array if imaginary part is small.

    Parameters
    ----------
    arr : array
    tol : float
        Absolute tolerance

    Returns
    -------
    arr : double or complex array
    """
    if tol is None:
        tol = {0:feps*1e3, 1:eps*1e6}[_array_precision[arr.dtype.char]]
    if (arr.dtype.char in ['F', 'D','G']) and \
       np.allclose(arr.imag, 0.0, atol=tol):
        arr = arr.real
    return arr

def cosm(A):
    """Compute the matrix cosine.

    This routine uses expm to compute the matrix exponentials.

    Parameters
    ----------
    A : array, shape(M,M)

    Returns
    -------
    cosA : array, shape(M,M)
        Matrix cosine of A

    """
    A = asarray(A)
    if A.dtype.char not in ['F','D','G']:
        return expm(1j*A).real
    else:
        return 0.5*(expm(1j*A) + expm(-1j*A))


def sinm(A):
    """Compute the matrix sine.

    This routine uses expm to compute the matrix exponentials.

    Parameters
    ----------
    A : array, shape(M,M)

    Returns
    -------
    sinA : array, shape(M,M)
        Matrix cosine of A

    """
    A = asarray(A)
    if A.dtype.char not in ['F','D','G']:
        return expm(1j*A).imag
    else:
        return -0.5j*(expm(1j*A) - expm(-1j*A))

def tanm(A):
    """Compute the matrix tangent.

    This routine uses expm to compute the matrix exponentials.

    Parameters
    ----------
    A : array, shape(M,M)

    Returns
    -------
    tanA : array, shape(M,M)
        Matrix tangent of A

    """
    A = asarray(A)
    if A.dtype.char not in ['F','D','G']:
        return toreal(solve(cosm(A), sinm(A)))
    else:
        return solve(cosm(A), sinm(A))

def coshm(A):
    """Compute the hyperbolic matrix cosine.

    This routine uses expm to compute the matrix exponentials.

    Parameters
    ----------
    A : array, shape(M,M)

    Returns
    -------
    coshA : array, shape(M,M)
        Hyperbolic matrix cosine of A

    """
    A = asarray(A)
    if A.dtype.char not in ['F','D','G']:
        return toreal(0.5*(expm(A) + expm(-A)))
    else:
        return 0.5*(expm(A) + expm(-A))

def sinhm(A):
    """Compute the hyperbolic matrix sine.

    This routine uses expm to compute the matrix exponentials.

    Parameters
    ----------
    A : array, shape(M,M)

    Returns
    -------
    sinhA : array, shape(M,M)
        Hyperbolic matrix sine of A

    """
    A = asarray(A)
    if A.dtype.char not in ['F','D']:
        return toreal(0.5*(expm(A) - expm(-A)))
    else:
        return 0.5*(expm(A) - expm(-A))

def tanhm(A):
    """Compute the hyperbolic matrix tangent.

    This routine uses expm to compute the matrix exponentials.

    Parameters
    ----------
    A : array, shape(M,M)

    Returns
    -------
    tanhA : array, shape(M,M)
        Hyperbolic matrix tangent of A

    """
    A = asarray(A)
    if A.dtype.char not in ['F','D']:
        return toreal(solve(coshm(A), sinhm(A)))
    else:
        return solve(coshm(A), sinhm(A))

def funm(A, func, disp=True):
    """Evaluate a matrix function specified by a callable.

    Returns the value of matrix-valued function f at A. The function f
    is an extension of the scalar-valued function func to matrices.

    Parameters
    ----------
    A : array, shape(M,M)
        Matrix at which to evaluate the function
    func : callable
        Callable object that evaluates a scalar function f.
        Must be vectorized (eg. using vectorize).
    disp : boolean
        Print warning if error in the result is estimated large
        instead of returning estimated error. (Default: True)

    Returns
    -------
    fA : array, shape(M,M)
        Value of the matrix function specified by func evaluated at A

    (if disp == False)
    errest : float
        1-norm of the estimated error, ||err||_1 / ||A||_1

    """
    # Perform Shur decomposition (lapack ?gees)
    A = asarray(A)
    if len(A.shape)!=2:
        raise ValueError("Non-matrix input to matrix function.")
    if A.dtype.char in ['F', 'D', 'G']:
        cmplx_type = 1
    else:
        cmplx_type = 0
    T, Z = schur(A)
    T, Z = rsf2csf(T,Z)
    n,n = T.shape
    F = diag(func(diag(T)))  # apply function to diagonal elements
    F = F.astype(T.dtype.char) # e.g. when F is real but T is complex

    minden = abs(T[0,0])

    # implement Algorithm 11.1.1 from Golub and Van Loan
    #                 "matrix Computations."
    for p in range(1,n):
        for i in range(1,n-p+1):
            j = i + p
            s = T[i-1,j-1] * (F[j-1,j-1] - F[i-1,i-1])
            ksl = slice(i,j-1)
            val = dot(T[i-1,ksl],F[ksl,j-1]) - dot(F[i-1,ksl],T[ksl,j-1])
            s = s + val
            den = T[j-1,j-1] - T[i-1,i-1]
            if den != 0.0:
                s = s / den
            F[i-1,j-1] = s
            minden = min(minden,abs(den))

    F = dot(dot(Z, F),transpose(conjugate(Z)))
    if not cmplx_type:
        F = toreal(F)

    tol = {0:feps, 1:eps}[_array_precision[F.dtype.char]]
    if minden == 0.0:
        minden = tol
    err = min(1, max(tol,(tol/minden)*norm(triu(T,1),1)))
    if product(ravel(logical_not(isfinite(F))),axis=0):
        err = Inf
    if disp:
        if err > 1000*tol:
            print "Result may be inaccurate, approximate err =", err
        return F
    else:
        return F, err

def logm(A, disp=True):
    """Compute matrix logarithm.

    The matrix logarithm is the inverse of expm: expm(logm(A)) == A

    Parameters
    ----------
    A : array, shape(M,M)
        Matrix whose logarithm to evaluate
    disp : boolean
        Print warning if error in the result is estimated large
        instead of returning estimated error. (Default: True)

    Returns
    -------
    logA : array, shape(M,M)
        Matrix logarithm of A

    (if disp == False)
    errest : float
        1-norm of the estimated error, ||err||_1 / ||A||_1

    """
    # Compute using general funm but then use better error estimator and
    #   make one step in improving estimate using a rotation matrix.
    A = mat(asarray(A))
    F, errest = funm(A,log,disp=0)
    errtol = 1000*eps
    # Only iterate if estimate of error is too large.
    if errest >= errtol:
        # Use better approximation of error
        errest = norm(expm(F)-A,1) / norm(A,1)
        if not isfinite(errest) or errest >= errtol:
            N,N = A.shape
            X,Y = ogrid[1:N+1,1:N+1]
            R = mat(orth(eye(N,dtype='d')+X+Y))
            F, dontcare = funm(R*A*R.H,log,disp=0)
            F = R.H*F*R
            if (norm(imag(F),1)<=1000*errtol*norm(F,1)):
                F = mat(real(F))
            E = mat(expm(F))
            temp = mat(solve(E.T,(E-A).T))
            F = F - temp.T
            errest = norm(expm(F)-A,1) / norm(A,1)
    if disp:
        if not isfinite(errest) or errest >= errtol:
            print "Result may be inaccurate, approximate err =", errest
        return F
    else:
        return F, errest

def signm(a, disp=True):
    """Matrix sign function.

    Extension of the scalar sign(x) to matrices.

    Parameters
    ----------
    A : array, shape(M,M)
        Matrix at which to evaluate the sign function
    disp : boolean
        Print warning if error in the result is estimated large
        instead of returning estimated error. (Default: True)

    Returns
    -------
    sgnA : array, shape(M,M)
        Value of the sign function at A

    (if disp == False)
    errest : float
        1-norm of the estimated error, ||err||_1 / ||A||_1

    Examples
    --------
    >>> from scipy.linalg import signm, eigvals
    >>> a = [[1,2,3], [1,2,1], [1,1,1]]
    >>> eigvals(a)
    array([ 4.12488542+0.j, -0.76155718+0.j,  0.63667176+0.j])
    >>> eigvals(signm(a))
    array([-1.+0.j,  1.+0.j,  1.+0.j])

    """
    def rounded_sign(x):
        rx = real(x)
        if rx.dtype.char=='f':
            c =  1e3*feps*amax(x)
        else:
            c =  1e3*eps*amax(x)
        return sign( (absolute(rx) > c) * rx )
    result,errest = funm(a, rounded_sign, disp=0)
    errtol = {0:1e3*feps, 1:1e3*eps}[_array_precision[result.dtype.char]]
    if errest < errtol:
        return result

    # Handle signm of defective matrices:

    # See "E.D.Denman and J.Leyva-Ramos, Appl.Math.Comp.,
    # 8:237-250,1981" for how to improve the following (currently a
    # rather naive) iteration process:

    a = asarray(a)
    #a = result # sometimes iteration converges faster but where??

    # Shifting to avoid zero eigenvalues. How to ensure that shifting does
    # not change the spectrum too much?
    vals = svd(a,compute_uv=0)
    max_sv = np.amax(vals)
    #min_nonzero_sv = vals[(vals>max_sv*errtol).tolist().count(1)-1]
    #c = 0.5/min_nonzero_sv
    c = 0.5/max_sv
    S0 = a + c*np.identity(a.shape[0])
    prev_errest = errest
    for i in range(100):
        iS0 = inv(S0)
        S0 = 0.5*(S0 + iS0)
        Pp=0.5*(dot(S0,S0)+S0)
        errest = norm(dot(Pp,Pp)-Pp,1)
        if errest < errtol or prev_errest==errest:
            break
        prev_errest = errest
    if disp:
        if not isfinite(errest) or errest >= errtol:
            print "Result may be inaccurate, approximate err =", errest
        return S0
    else:
        return S0, errest

def sqrtm(A, disp=True):
    """Matrix square root.

    Parameters
    ----------
    A : array, shape(M,M)
        Matrix whose square root to evaluate
    disp : boolean
        Print warning if error in the result is estimated large
        instead of returning estimated error. (Default: True)

    Returns
    -------
    sgnA : array, shape(M,M)
        Value of the sign function at A

    (if disp == False)
    errest : float
        Frobenius norm of the estimated error, ||err||_F / ||A||_F

    Notes
    -----
    Uses algorithm by Nicholas J. Higham

    """
    A = asarray(A)
    if len(A.shape)!=2:
        raise ValueError("Non-matrix input to matrix function.")
    T, Z = schur(A)
    T, Z = rsf2csf(T,Z)
    n,n = T.shape

    R = np.zeros((n,n),T.dtype.char)
    for j in range(n):
        R[j,j] = sqrt(T[j,j])
        for i in range(j-1,-1,-1):
            s = 0
            for k in range(i+1,j):
                s = s + R[i,k]*R[k,j]
            R[i,j] = (T[i,j] - s)/(R[i,i] + R[j,j])

    R, Z = all_mat(R,Z)
    X = (Z * R * Z.H)

    if disp:
        nzeig = np.any(diag(T)==0)
        if nzeig:
            print "Matrix is singular and may not have a square root."
        return X.A
    else:
        arg2 = norm(X*X - A,'fro')**2 / norm(A,'fro')
        return X.A, arg2