# publicscipy/scipy

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 `"""Schur decomposition functions."""import numpyfrom numpy import asarray_chkfinite, single# Local imports.import miscfrom misc import LinAlgError, _datacopiedfrom lapack import get_lapack_funcsfrom decomp import eigvals__all__ = ['schur', 'rsf2csf']_double_precision = ['i','l','d']def schur(a, output='real', lwork=None, overwrite_a=False, sort=None):    """Compute Schur decomposition of a matrix. The Schur decomposition is A = Z T Z^H where Z is unitary and T is either upper-triangular, or for real Schur decomposition (output='real'), quasi-upper triangular. In the quasi-triangular form, 2x2 blocks describing complex-valued eigenvalue pairs may extrude from the diagonal. Parameters ---------- a : array, shape (M, M) Matrix to decompose output : {'real', 'complex'} Construct the real or complex Schur decomposition (for real matrices). lwork : integer Work array size. If None or -1, it is automatically computed. overwrite_a : boolean Whether to overwrite data in a (may improve performance) sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'} Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given a eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). Alternatively, string parameters may be used: 'lhp' Left-hand plane (x.real < 0.0) 'rhp' Right-hand plane (x.real > 0.0) 'iuc' Inside the unit circle (x*x.conjugate() <= 1.0) 'ouc' Outside the unit circle (x*x.conjugate() > 1.0) Defaults to None (no sorting). Returns ------- T : array, shape (M, M) Schur form of A. It is real-valued for the real Schur decomposition. Z : array, shape (M, M) An unitary Schur transformation matrix for A. It is real-valued for the real Schur decomposition. sdim : integer If and only if sorting was requested, a third return value will contain the number of eigenvalues satisfying the sort condition. Raises ------ LinAlgError Error raised under three conditions: 1. The algorithm failed due to a failure of the QR algorithm to compute all eigenvalues 2. If eigenvalue sorting was requested, the eigenvalues could not be reordered due to a failure to separate eigenvalues, usually because of poor conditioning 3. If eigenvalue sorting was requested, roundoff errors caused the leading eigenvalues to no longer satisfy the sorting condition See also -------- rsf2csf : Convert real Schur form to complex Schur form """    if not output in ['real','complex','r','c']:        raise ValueError("argument must be 'real', or 'complex'")    a1 = asarray_chkfinite(a)    if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):        raise ValueError('expected square matrix')    typ = a1.dtype.char    if output in ['complex','c'] and typ not in ['F','D']:        if typ in _double_precision:            a1 = a1.astype('D')            typ = 'D'        else:            a1 = a1.astype('F')            typ = 'F'    overwrite_a = overwrite_a or (_datacopied(a1, a))    gees, = get_lapack_funcs(('gees',), (a1,))    if lwork is None or lwork == -1:        # get optimal work array        result = gees(lambda x: None, a1, lwork=-1)        lwork = result[-2][0].real.astype(numpy.int)    if sort is None:        sort_t = 0        sfunction = lambda x: None    else:        sort_t = 1        if callable(sort):            sfunction = sort        elif sort == 'lhp':            sfunction = lambda x: (numpy.real(x) < 0.0)        elif sort == 'rhp':            sfunction = lambda x: (numpy.real(x) >= 0.0)        elif sort == 'iuc':            sfunction = lambda x: (abs(x) <= 1.0)        elif sort == 'ouc':            sfunction = lambda x: (abs(x) > 1.0)        else:            raise ValueError("sort parameter must be None, a callable, or " +                "one of ('lhp','rhp','iuc','ouc')")    result = gees(sfunction, a1, lwork=lwork, overwrite_a=overwrite_a,        sort_t=sort_t)    info = result[-1]    if info < 0:        raise ValueError('illegal value in %d-th argument of internal gees'                                                                    % -info)    elif info == a1.shape[0] + 1:        raise LinAlgError('Eigenvalues could not be separated for reordering.')    elif info == a1.shape[0] + 2:        raise LinAlgError('Leading eigenvalues do not satisfy sort condition.')    elif info > 0:        raise LinAlgError("Schur form not found. Possibly ill-conditioned.")    if sort_t == 0:        return result[0], result[-3]    else:        return result[0], result[-3], result[1]eps = numpy.finfo(float).epsfeps = numpy.finfo(single).eps_array_kind = {'b':0, 'h':0, 'B': 0, 'i':0, 'l': 0, 'f': 0, 'd': 0, 'F': 1, 'D': 1}_array_precision = {'i': 1, 'l': 1, 'f': 0, 'd': 1, 'F': 0, 'D': 1}_array_type = [['f', 'd'], ['F', 'D']]def _commonType(*arrays):    kind = 0    precision = 0    for a in arrays:        t = a.dtype.char        kind = max(kind, _array_kind[t])        precision = max(precision, _array_precision[t])    return _array_type[kind][precision]def _castCopy(type, *arrays):    cast_arrays = ()    for a in arrays:        if a.dtype.char == type:            cast_arrays = cast_arrays + (a.copy(),)        else:            cast_arrays = cast_arrays + (a.astype(type),)    if len(cast_arrays) == 1:        return cast_arrays[0]    else:        return cast_arraysdef rsf2csf(T, Z):    """Convert real Schur form to complex Schur form. Convert a quasi-diagonal real-valued Schur form to the upper triangular complex-valued Schur form. Parameters ---------- T : array, shape (M, M) Real Schur form of the original matrix Z : array, shape (M, M) Schur transformation matrix Returns ------- T : array, shape (M, M) Complex Schur form of the original matrix Z : array, shape (M, M) Schur transformation matrix corresponding to the complex form See also -------- schur : Schur decompose a matrix """    Z, T = map(asarray_chkfinite, (Z, T))    if len(Z.shape) != 2 or Z.shape[0] != Z.shape[1]:        raise ValueError("matrix must be square.")    if len(T.shape) != 2 or T.shape[0] != T.shape[1]:        raise ValueError("matrix must be square.")    if T.shape[0] != Z.shape[0]:        raise ValueError("matrices must be same dimension.")    N = T.shape[0]    arr = numpy.array    t = _commonType(Z, T, arr([3.0],'F'))    Z, T = _castCopy(t, Z, T)    conj = numpy.conj    dot = numpy.dot    r_ = numpy.r_    transp = numpy.transpose    for m in range(N-1, 0, -1):        if abs(T[m,m-1]) > eps*(abs(T[m-1,m-1]) + abs(T[m,m])):            k = slice(m-1, m+1)            mu = eigvals(T[k,k]) - T[m,m]            r = misc.norm([mu[0], T[m,m-1]])            c = mu[0] / r            s = T[m,m-1] / r            G = r_[arr([[conj(c), s]], dtype=t), arr([[-s, c]], dtype=t)]            Gc = conj(transp(G))            j = slice(m-1, N)            T[k,j] = dot(G, T[k,j])            i = slice(0, m+1)            T[i,k] = dot(T[i,k], Gc)            i = slice(0, N)            Z[i,k] = dot(Z[i,k], Gc)        T[m,m-1] = 0.0;    return T, Z`
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