svd                   package:base                   R Documentation

_S_i_n_g_u_l_a_r _V_a_l_u_e _D_e_c_o_m_p_o_s_i_t_i_o_n _o_f _a _M_a_t_r_i_x

_D_e_s_c_r_i_p_t_i_o_n:

     Compute the singular-value decomposition of a rectangular matrix.

_U_s_a_g_e:

     svd(x, nu = min(n, p), nv = min(n, p))
     La.svd(x, nu = min(n, p), nv = min(n, p))

_A_r_g_u_m_e_n_t_s:

       x: a matrix whose SVD decomposition is to be computed.

      nu: the number of left  singular vectors to be computed. This
          must be one of `0', `nrow(x)' and `ncol(x)'.

      nv: the number of right singular vectors to be computed. This
          must be one of `0' and `ncol(x)'.

_D_e_t_a_i_l_s:

     The singular value decomposition plays an important role in many
     statistical techniques.

     `svd' provides an interface to the LINPACK routine DSVDC. `La.svd'
     provides an interface to the LAPACK routine DGESVD.

     `La.svd' is preferred to `svd' for new projects, but it is not an
     exact replacement as it returns the transpose of the right
     singular vector matrix, and the signs of the singular vectors may
     differ from those given by `svd'.

     Both functions handle complex matrices via LAPACK routine ZGESVD.

_V_a_l_u_e:

     The SVD decomposition of the matrix as computed by LINPACK,

                             X = U D V',

     where U and V are orthogonal, V' means V transposed, and D is a
     diagonal matrix with the singular values D[i,i].  Equivalently, D
     = U' X V, which is verified in the examples, below.

     The components in the returned value correspond directly to the
     values returned by DSVDC. 

       d: a vector containing the singular values of `x'.

       u: a matrix whose columns contain the left singular vectors of
          `x'.

       v: a matrix whose columns contain the right singular vectors of
          `x'.


     For `La.svd' the return value replaces `v' by `vt', the
     (conjugated if complex) transpose of `v'.

_R_e_f_e_r_e_n_c_e_s:

     Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W.
     (1978) LINPACK Users Guide.  Philadelphia: SIAM Publications.

     Anderson. E. and ten others (1995) LAPACK Users' Guide. Second
     Edition. SIAM.

_S_e_e _A_l_s_o:

     `eigen', `qr'.

_E_x_a_m_p_l_e_s:

     hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
     str(X <- hilbert(9)[,1:6])
     str(s <- svd(X))
     Eps <- 100 * .Machine$double.eps

     D <- diag(s$d)
     stopifnot(abs(X - s$u %*% D %*% t(s$v)) < Eps)#  X = U D V'
     stopifnot(abs(D - t(s$u) %*% X %*% s$v) < Eps)#  D = U' X V

     X <- cbind(1, 1:7)
     str(s <- svd(X)); D <- diag(s$d)
     stopifnot(abs(X - s$u %*% D %*% t(s$v)) < Eps)#  X = U D V'
     stopifnot(abs(D - t(s$u) %*% X %*% s$v) < Eps)#  D = U' X V

