(octave.info)Functions of a Matrix

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18.4 Functions of a Matrix
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 -- : expm (A)
     Return the exponential of a matrix.

     The matrix exponential is defined as the infinite Taylor series

          expm (A) = I + A + A^2/2! + A^3/3! + ...

     However, the Taylor series is _not_ the way to compute the matrix
     exponential; see Moler and Van Loan, ‘Nineteen Dubious Ways to
     Compute the Exponential of a Matrix’, SIAM Review, 1978.  This
     routine uses Ward’s diagonal Padé approximation method with three
     step preconditioning (SIAM Journal on Numerical Analysis, 1977).
     Diagonal Padé approximations are rational polynomials of matrices

               -1
          D (A)   N (A)

     whose Taylor series matches the first ‘2q+1’ terms of the Taylor
     series above; direct evaluation of the Taylor series (with the same
     preconditioning steps) may be desirable in lieu of the Padé
     approximation when ‘Dq(A)’ is ill-conditioned.

     See also: Note: logm, Note: sqrtm.

 -- : S = logm (A)
 -- : S = logm (A, OPT_ITERS)
 -- : [S, ITERS] = logm (...)
     Compute the matrix logarithm of the square matrix A.

     The implementation utilizes a Padé approximant and the identity

          logm (A) = 2^k * logm (A^(1 / 2^k))

     The optional input OPT_ITERS is the maximum number of square roots
     to compute and defaults to 100.

     The optional output ITERS is the number of square roots actually
     computed.

     See also: Note: expm, Note: sqrtm.

 -- : S = sqrtm (A)
 -- : [S, ERROR_ESTIMATE] = sqrtm (A)
     Compute the matrix square root of the square matrix A.

     Ref: N.J. Higham.  ‘A New sqrtm for MATLAB’.  Numerical Analysis
     Report No.  336, Manchester Centre for Computational Mathematics,
     Manchester, England, January 1999.

     See also: Note: expm, Note: logm.

 -- : kron (A, B)
 -- : kron (A1, A2, ...)
     Form the Kronecker product of two or more matrices.

     This is defined block by block as

          x = [ a(i,j)*b ]

     For example:

          kron (1:4, ones (3, 1))
               ⇒  1  2  3  4
                   1  2  3  4
                   1  2  3  4

     If there are more than two input arguments A1, A2, ..., AN the
     Kronecker product is computed as

          kron (kron (A1, A2), ..., AN)

     Since the Kronecker product is associative, this is well-defined.

 -- : blkmm (A, B)
     Compute products of matrix blocks.

     The blocks are given as 2-dimensional subarrays of the arrays A, B.
     The size of A must have the form ‘[m,k,...]’ and size of B must be
     ‘[k,n,...]’.  The result is then of size ‘[m,n,...]’ and is
     computed as follows:

          for i = 1:prod (size (A)(3:end))
            C(:,:,i) = A(:,:,i) * B(:,:,i)
          endfor

 -- : X = sylvester (A, B, C)
     Solve the Sylvester equation.

     The Sylvester equation is defined as:

          A X + X B = C

     The solution is computed using standard LAPACK subroutines.

     For example:

          sylvester ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
             ⇒ [ 0.50000, 0.66667; 0.66667, 0.50000 ]