Python linalg.eigh函数代码示例

def eigenvectors(h,nk=10):
  import scipy.linalg as lg
  from scipy.sparse import csc_matrix as csc
  if h.dimensionality==0:
    vv = lg.eigh(h.intra)
    vecs = [v for v in vv[1].transpose()]
    return vv[0],vecs
  elif h.dimensionality>0:
    f = h.get_hk_gen()
    if h.dimensionality==1: kp = np.linspace(0.,1.0,nk,endpoint=False)
    if h.dimensionality==2: 
      kp = []
      for k1 in np.linspace(0.,1.0,nk,endpoint=False):
        for k2 in np.linspace(0.,1.0,nk,endpoint=False):
          kp.append([k1,k2]) # store
    eigvecs = np.zeros((len(kp)*h.intra.shape[0],h.intra.shape[0]),dtype=np.complex) # eigenvectors
    eigvals = np.zeros((len(kp)*h.intra.shape[0])) # eigenvalues
    iv = 0
    for k in kp: # loop over kpoints
      hk = f(k)  # kdependent hamiltonian
      vv = lg.eigh(hk) # diagonalize k hamiltonian
      for (e,v) in zip(vv[0],vv[1].transpose()):
        eigvecs[iv] = v.copy()
        eigvals[iv] = e.copy()
        iv += 1
#      eigvals += vv[0].tolist() # store eigenvalues
#      vecs = [v for v in vv[1].transpose()]
#      eigvecs += vecs # store eigenvectors 
    return eigvals,eigvecs
  else:
    raise

    def set(self,gse):
        '''
        Set the Lambda matrix, Q matrix and QT matrix of the block.

        Parameters
        ----------
        gse : number
            The groundstate energy.
        '''
        sign=self.controllers['sign']
        if self.method=='S':
            lczs,Qs=self.controllers['lczs'],self.controllers['Qs']
            self.data['niters']=np.zeros(Qs.shape[0],dtype=np.int32)
            self.data['Lambdas']=np.zeros((Qs.shape[0],Qs.shape[2]),dtype=np.float64)
            self.data['Qs']=np.zeros(Qs.shape,dtype=Qs.dtype)
            self.data['QTs']=np.zeros((Qs.shape[0],Qs.shape[2]),dtype=Qs.dtype)
            for i,(lcz,Q) in enumerate(zip(lczs,Qs)):
                if lcz.niter>0:
                    E,V=sl.eigh(lcz.T,eigvals_only=False)
                    self.data['niters'][i]=lcz.niter
                    self.data['Lambdas'][i,0:lcz.niter]=sign*(E-gse)
                    self.data['Qs'][i,:,0:lcz.niter]=Q[:,0:lcz.niter].dot(V)
                    self.data['QTs'][i,0:lcz.niter]=lcz.P[0,0]*V[0,:].conjugate()
        else:
            lanczos=self.controllers['lanczos']
            E,V=sl.eigh(lanczos.T,eigvals_only=False)
            self.data['Lambda']=sign*(E-gse)
            self.data['Q']=lanczos.P[:min(lanczos.nv0,lanczos.niter),:].T.conjugate().dot(V[:min(lanczos.nv0,lanczos.niter),:])
            self.data['QT']=HM.dagger(self.data['Q'])

def jitEigh(A,maxTries=10,warning=True):
    """
    Do a Eigenvalue Decompsition with Jitter,

    works as jitChol
    """
    warning = True
    jitter = 0
    i = 0

    while(True):
        if jitter == 0:
            jitter = abs(SP.trace(A))/A.shape[0]*1e-6
            S,U = linalg.eigh(A)

        else:
            if warning:
                # pdb.set_trace()
		# plt.figure()
		# plt.imshow(A, interpolation="nearest")
		# plt.colorbar()
		# plt.show()
                logging.error("Adding jitter of %f in jitEigh()." % jitter)
            S,U = linalg.eigh(A+jitter*SP.eye(A.shape[0]))

        if S.min()>1E-10:
            return S,U
            
        if i<maxTries:
            jitter = jitter*10
        i += 1
            
    raise linalg.LinAlgError, "Matrix non positive definite, jitter of " +  str(jitter) + " added but failed after " + str(i) + " trials."

def geneigh(A,B,tol=1e-12):
    """
    Solves the generalized eigenvalue problem also in the case where A and B share a common
    null-space. The eigenvalues corresponding to the null-space are given a Nan value.
    The null-space is defined with the tolereance tol.
    """
    # first check if there is a null-space issue
    if lg.matrix_rank(B,tol)==np.shape(B)[0]:
        return eigh(A,B)
    # first diagonalize the overlap matrix B
    Be,Bv=eigh(B)
    # rewrite the A matrix in the B-matrix eigenspace
    At=np.dot(np.conj(Bv.T),np.dot(A,Bv))
    Bt=np.diag(Be)
    # detect shared null-space. that is given by the first n null eigenvalues of B
    try:
        idx=next(i for i,v in enumerate(Be) if v>tol)
    except StopIteration:
        raise(RuntimeError('geneigh: Rank of B < B.shape[0] but null-space could not be found!'))
    # check that the B matrix null-space is shared by A.
    m=np.amax(abs(At[0:idx,:].flatten()))
    if m>tol:
        warnings.warn('Maximum non-diagonal element in A written in B null-space is bigger than the tolerance \''+str(tol)+'\'.',UserWarning)
    # diagonalize the non-null-space part of the problem
    Et,Vt=eigh(At[idx:,idx:],Bt[idx:,idx:])
    # define Ut, the change of basis in the non-truncated space
    Ut=np.zeros(np.shape(A),A.dtype)
    Ut[0:idx,0:idx]=np.eye(idx)
    Ut[idx:,idx:]=Vt
    U=np.dot(Bv,Ut)
    E=np.concatenate((float('NaN')*np.ones(idx),Et))
    return E,U

def TBAEB(engine,app):
    nmatrix=len(engine.generators['h'].table)
    if app.path!=None:
        key=app.path.mesh.keys()[0]
        result=zeros((app.path.rank[key],nmatrix+1))
        if len(app.path.mesh[key].shape)==1:
            result[:,0]=app.path.mesh[key]
        else:
            result[:,0]=array(xrange(app.path.rank[key]))
        for i,parameter in enumerate(list(app.path.mesh[key])):
            result[i,1:]=eigh(engine.matrix(**{key:parameter}),eigvals_only=True)
    else:
        result=zeros((2,nmatrix+1))
        result[:,0]=array(xrange(2))
        result[0,1:]=eigh(engine.matrix(),eigvals_only=True)
        result[1,1:]=result[0,1:]
    if app.save_data:
        savetxt(engine.dout+'/'+engine.name.full+'_EB.dat',result)
    if app.plot:
        plt.title(engine.name.full+'_EB')
        plt.plot(result[:,0],result[:,1:])
        if app.show:
            plt.show()
        else:
            plt.savefig(engine.dout+'/'+engine.name.full+'_EB.png')
        plt.close()

def _eval_cov_learner(X, train_ix, test_ix, model_prec, model_cov,
                      cov_learner, ips_flag=True):
    X_train = X[train_ix, ...]
    alpha_max_ = alpha_max(X_train)
    if model_prec is None and model_cov is None:
        X_test = X[test_ix, ...]
    elif model_cov is None:
        eigvals, eigvecs = linalg.eigh(model_prec)
        X_test = np.diag(1. / np.sqrt(eigvals)).dot(eigvecs.T)
    else:
        eigvals, eigvecs = linalg.eigh(model_prec)
        X_test = np.diag(np.sqrt(eigvals)).dot(eigvecs.T)
    cov_learner_ = clone(cov_learner)
    cov_learner_.__setattr__('alpha', cov_learner_.alpha * alpha_max_)
    if not ips_flag:
        score = cov_learner_.fit(X_train).score(X_test)
    elif cov_learner.score_norm != "ell0":
        # dual split variable contains exact zeros!
        aux_prec = cov_learner_.fit(X_train).auxiliary_prec_
        mask = np.abs(aux_prec) > machine_eps(0.)
        ips = IPS(support=mask, score_norm=cov_learner_.score_norm)
        score = ips.fit(X_train).score(X_test)
    else:
        raise ValueError('ell0 scoring in CV_loop and IPS are incompatible')

    # make scores maximal at optimum
    if cov_learner_.score_norm not in {'loglikelihood', None}:
        score *= -1.
    return score

def test_hermitian():
    np.random.seed(1234)

    sizes = [3, 10, 50]
    ks = [1, 3, 10, 50]
    gens = [True, False]

    for size, k, gen in itertools.product(sizes, ks, gens):
        if k > size:
            continue

        H = np.random.rand(size, size) + 1.j * np.random.rand(size, size)
        H = 10 * np.eye(size) + H + H.T.conj()

        X = np.random.rand(size, k)

        if not gen:
            B = np.eye(size)
            w, v = lobpcg(H, X, maxiter=5000)
            w0, v0 = eigh(H)
        else:
            B = np.random.rand(size, size) + 1.j * np.random.rand(size, size)
            B = 10 * np.eye(size) + B.dot(B.T.conj())
            w, v = lobpcg(H, X, B, maxiter=5000)
            w0, v0 = eigh(H, B)

        for wx, vx in zip(w, v.T):
            # Check eigenvector
            assert_allclose(np.linalg.norm(H.dot(vx) - B.dot(vx) * wx) / np.linalg.norm(H.dot(vx)),
                            0, atol=5e-4, rtol=0)

            # Compare eigenvalues
            j = np.argmin(abs(w0 - wx))
            assert_allclose(wx, w0[j], rtol=1e-4)

def fitPairwiseModel(Y,XX=None,S_XX=None,U_XX=None,verbose=False):
    N,P = Y.shape
    """ initilizes parameters """
    RV = fitSingleTraitModel(Y,XX=XX,S_XX=S_XX,U_XX=U_XX,verbose=verbose)
    Cg = covariance.freeform(2)
    Cn = covariance.freeform(2)
    gp = gp2kronSum(mean(Y[:,0:2]),Cg,Cn,XX=XX,S_XX=S_XX,U_XX=U_XX)
    conv2 = SP.ones((P,P),dtype=bool)
    rho_g = SP.ones((P,P))
    rho_n = SP.ones((P,P))
    for p1 in range(P):
        for p2 in range(p1):
            if verbose:
                print '.. fitting correlation (%d,%d)'%(p1,p2)
            gp.setY(Y[:,[p1,p2]])
            Cg_params0 = SP.array([SP.sqrt(RV['varST'][p1,0]),1e-6*SP.randn(),SP.sqrt(RV['varST'][p2,0])])
            Cn_params0 = SP.array([SP.sqrt(RV['varST'][p1,1]),1e-6*SP.randn(),SP.sqrt(RV['varST'][p2,1])])
            params0 = {'Cg':Cg_params0,'Cn':Cn_params0}
            conv2[p1,p2],info = OPT.opt_hyper(gp,params0,factr=1e3)
            rho_g[p1,p2] = Cg.K()[0,1]/SP.sqrt(Cg.K().diagonal().prod())
            rho_n[p1,p2] = Cn.K()[0,1]/SP.sqrt(Cn.K().diagonal().prod())
            conv2[p2,p1] = conv2[p1,p2]; rho_g[p2,p1] = rho_g[p1,p2]; rho_n[p2,p1] = rho_n[p1,p2]
    RV['Cg0'] = rho_g*SP.dot(SP.sqrt(RV['varST'][:,0:1]),SP.sqrt(RV['varST'][:,0:1].T))
    RV['Cn0'] = rho_n*SP.dot(SP.sqrt(RV['varST'][:,1:2]),SP.sqrt(RV['varST'][:,1:2].T))
    RV['conv2'] = conv2
    #3. regularizes covariance matrices
    offset_g = abs(SP.minimum(LA.eigh(RV['Cg0'])[0].min(),0))+1e-4
    offset_n = abs(SP.minimum(LA.eigh(RV['Cn0'])[0].min(),0))+1e-4
    RV['Cg0_reg'] = RV['Cg0']+offset_g*SP.eye(P)
    RV['Cn0_reg'] = RV['Cn0']+offset_n*SP.eye(P)
    RV['params0_Cg']=LA.cholesky(RV['Cg0_reg'])[SP.tril_indices(P)]
    RV['params0_Cn']=LA.cholesky(RV['Cn0_reg'])[SP.tril_indices(P)]
    return RV

def pca_whiten(x2d, n_comp, verbose=True):
    """ data Whitening
    *Input
    x2d : 2d data matrix of observations by variables
    n_comp: Number of components to retain
    *Output
    Xwhite : Whitened X
    white : whitening matrix (Xwhite = np.dot(white,X))
    dewhite : dewhitening matrix (X = np.dot(dewhite,Xwhite))
    """
    x2d_demean = x2d - x2d.mean(axis=1).reshape((-1, 1))
    samples, features = x2d_demean.shape
    M = min((samples, features))
    if samples > features:
        cov = dot(x2d_demean.T, x2d_demean) / (x2d.shape[0] - 1)
        w, v = eigh(cov, eigvals=(M-n_comp, M-1))
        D, Di = diagsqrts(w)
        u = dot(dot(x2d_demean,v),Di)
        x_white = v.T
        white = dot(Di, u.T)
        dewhite = dot(u, D)
    else:
        cov = dot(x2d_demean, x2d_demean.T) / (x2d.shape[1] - 1)
        w, u = eigh(cov, eigvals=(M-n_comp, M-1))
        D, Di = diagsqrts(w)        
        white = dot(Di, u.T)
        x_white = dot(white, x2d_demean)
        dewhite = dot(u, D)
    return (x_white, white, dewhite)

 def getEig1(self):
     '''
     '''
     if self.eig1 is None:
         #compute eig1
         if self.K1 is not None:
             if self.K1rot is None:
                 self.K1rot = rotSymm(self.K1, eig = self.eig0, exponent = -0.5, gamma=self.gamma0,delta = self.delta,forceSymm = False)
                 self.K1rot = rotSymm(self.K1rot.T, eig = self.eig0, exponent = -0.5, gamma=self.gamma0,delta = self.delta,forceSymm = False)
             self.eig1 = LA.eigh(self.K1rot)
         elif self.G1 is not None:
             [N,k] = self.G1.shape
             if self.G1rot is None:
                 self.G1rot = rotSymm(self.G1, eig = self.eig0, exponent = -0.5, gamma=self.gamma0,delta = self.delta,forceSymm = False)
             
             try:
                 [U,S,V] = LA.svd(self.G1rot,full_matrices = False)
                 self.eig1 = [S*S,U]
             except LA.LinAlgError:  # revert to Eigenvalue decomposition
                 print "Got SVD exception, trying eigenvalue decomposition of square of G. Note that this is a little bit less accurate"
                 [S_,V_] = LA.eigh(self.G1rot.T.dot(self.G1rot))
                 S_nonz=(S_>0.0)
                 S1 = S_[S_nonz]
                 U1=self.G1rot.dot(V_[:,S_nonz]/SP.sqrt(S1))
                 self.eig1=[S1,U1]
     return self.eig1

def geneigh(A,B,tol=1e-12):
    """
    Solves the generalized eigenvalue problem also in the case where A and B share a common
    null-space. The eigenvalues corresponding to the null-space are given a Nan value.
    The null-space is defined with the tolereance tol.
    """
    # first check if there is a null-space issue
    if matrix_rank(B,tol)==shape(B)[0]:
        return eigh(A,B)
    # first diagonalize the overlap matrix B
    Be,Bv=eigh(B)
    # rewrite the A matrix in the B-matrix eigenspace
    At=dot(conj(Bv.T),dot(A,Bv))
    Bt=diag(Be)
    # detect shared null-space. that is given by the first n null eigenvalues of B
    idx=find(Be>tol)
    idx=idx[0]
    # check that the B matrix null-space is shared by A.
    m=amax(abs(At[0:idx,:].flatten()))
    if m>tol:
        warnings.warn('Maximum non-diagonal element in A written in B null-space is bigger than the tolerance \''+str(tol)+'\'.',UserWarning)
    # diagonalize the non-null-space part of the problem
    Et,Vt=eigh(At[idx:,idx:],Bt[idx:,idx:])
    # define Ut, the change of basis in the non-truncated space
    Ut=zeros(shape(A),A.dtype)
    Ut[0:idx,0:idx]=eye(idx)
    Ut[idx:,idx:]=Vt
    U=dot(Bv,Ut)
    E=append(float('NaN')*ones(idx),Et)
    return E,U

    def _fit(self, cov_a, cov_b):
        """Aux Function (modifies cov_a and cov_b in-place)."""
        cov_a /= np.trace(cov_a)
        cov_b /= np.trace(cov_b)
        # computes the eigen values
        lambda_, u = linalg.eigh(cov_a + cov_b)
        # sort them
        ind = np.argsort(lambda_)[::-1]
        lambda2_ = lambda_[ind]

        u = u[:, ind]
        p = np.dot(np.sqrt(linalg.pinv(np.diag(lambda2_))), u.T)

        # Compute the generalized eigen value problem
        w_a = np.dot(np.dot(p, cov_a), p.T)
        w_b = np.dot(np.dot(p, cov_b), p.T)
        # and solve it
        vals, vecs = linalg.eigh(w_a, w_b)
        # sort vectors by discriminative power using eigen values
        ind = np.argsort(np.maximum(vals, 1.0 / vals))[::-1]
        vecs = vecs[:, ind]
        # and project
        w = np.dot(vecs.T, p)

        self.filters_ = w
        self.patterns_ = linalg.pinv(w).T

def numpy_matrix_operator_with_arrays_and_products_factory(dim_source, dim_range, count_source, count_range, seed,
                                                           source_id=None, range_id=None):
    from scipy.linalg import eigh
    op, _, U, V = numpy_matrix_operator_with_arrays_factory(dim_source, dim_range, count_source, count_range, seed,
                                                            source_id=source_id, range_id=range_id)
    if dim_source > 0:
        while True:
            sp = np.random.random((dim_source, dim_source))
            sp = sp.T.dot(sp)
            evals = eigh(sp, eigvals_only=True)
            if np.min(evals) > 1e-6:
                break
        sp = NumpyMatrixOperator(sp, source_id=source_id, range_id=source_id)
    else:
        sp = NumpyMatrixOperator(np.zeros((0, 0)), source_id=source_id, range_id=source_id)
    if dim_range > 0:
        while True:
            rp = np.random.random((dim_range, dim_range))
            rp = rp.T.dot(rp)
            evals = eigh(rp, eigvals_only=True)
            if np.min(evals) > 1e-6:
                break
        rp = NumpyMatrixOperator(rp, source_id=range_id, range_id=range_id)
    else:
        rp = NumpyMatrixOperator(np.zeros((0, 0)), source_id=range_id, range_id=range_id)
    return op, None, U, V, sp, rp

 def simulate(self,standardize=True):
     self._update_cache()
     RV = SP.zeros((self.N,self.P))
     # region
     Z = SP.randn(self.S,self.P)
     Sc,Uc = LA.eigh(self.Cr.K())
     Sc[Sc<1e-9] = 0
     USh_c = Uc*Sc[SP.newaxis,:]**0.5 
     RV += SP.dot(SP.dot(self.Xr,Z),USh_c.T)
     # background
     Z = SP.randn(self.N,self.P)
     USh_r = self.cache['Lr'].T*self.cache['Srstar'][SP.newaxis,:]**0.5
     Sc,Uc = LA.eigh(self.Cg.K())
     Sc[Sc<1e-9] = 0
     USh_c = Uc*Sc[SP.newaxis,:]**0.5
     RV += SP.dot(SP.dot(USh_r,Z),USh_c.T)
     # noise
     Z = SP.randn(self.N,self.P)
     Sc,Uc = LA.eigh(self.Cn.K())
     Sc[Sc<1e-9] = 0
     USh_c = Uc*Sc[SP.newaxis,:]**0.5 
     RV += SP.dot(Z,USh_c.T)
     # standardize
     if standardize:
         RV-=RV.mean(0)
         RV/=RV.std(0) 
     return RV

def cca(X, Y, k, SMALL=1e-5):
    """Standard CCA.

    Views _X_ and _Y_ are per *row*.

    For an explanation of the algorithm, 
    see Section 6.4 and 6.5 in
    Kernel Methods for Pattern Analysis.
    """
    n, dx = X.shape
    C = np.cov(X.T, Y.T)
    Cxy = C[:dx, dx:]
    Cxx = C[:dx, :dx] + SMALL * np.eye(dx)
    Cyy = C[dx:, dx:] + SMALL * np.eye(Y.shape[1])

    # Do not use la.sqrtm.
    # This can be done by hand...
    xeval, xevec = la.eigh(Cxx)
    yeval, yevec = la.eigh(Cyy)

    # ... because the inverses are then simple
    isqrtx = np.dot(xevec, (xevec/np.sqrt(xeval)).T)
    isqrty = np.dot(yevec, (yevec/np.sqrt(yeval)).T)

    tmp = np.dot(isqrtx, Cxy)
    tmp = np.dot(tmp, isqrty)
    [U, S, V] = la.svd(tmp, full_matrices=False)

    ccX = np.dot(isqrtx, U[:,:k])
    ccY = np.dot(isqrty, V[:k].T)
    
    return ccX, ccY