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revision 808 by gross, Thu Aug 10 01:12:16 2006 UTC revision 809 by gross, Sun Aug 13 22:10:03 2006 UTC
# Line 803  returns the non--symmetric part of \var{ Line 803  returns the non--symmetric part of \var{
803  \begin{funcdesc}{inverse}{a}  \begin{funcdesc}{inverse}{a}
804  return the inverse of \var{a}. This is  return the inverse of \var{a}. This is
805  \begin{equation}  \begin{equation}
806  \code{matrixmult(inverse(a),a)=kronecker(d)}  \code{matrix_mult(inverse(a),a)=kronecker(d)}
807  \end{equation}  \end{equation}
808  if \var{a} has shape \code{(d,d)}. The current implementation is restricted to arguments of shape  if \var{a} has shape \code{(d,d)}. The current implementation is restricted to arguments of shape
809  \code{(2,2)} and \code{(3,3)}.  \code{(2,2)} and \code{(3,3)}.
# Line 811  if \var{a} has shape \code{(d,d)}. The c Line 811  if \var{a} has shape \code{(d,d)}. The c
811  \begin{funcdesc}{eigenvalues}{a}  \begin{funcdesc}{eigenvalues}{a}
812  return the eigenvalues of \var{a}. This is  return the eigenvalues of \var{a}. This is
813  \begin{equation}  \begin{equation}
814  \code{matrixmult(a,V)=e[i]*V}  \code{matrix_mult(a,V)=e[i]*V}
815  \end{equation}  \end{equation}
816  where \code{e=eigenvalues(a)} and \var{V} is suitable non--zero vector \var{V}.  where \code{e=eigenvalues(a)} and \var{V} is suitable non--zero vector \var{V}.
817  The eigenvalues are ordered in increasing size.  The eigenvalues are ordered in increasing size.
# Line 822  The current implementation is restricted Line 822  The current implementation is restricted
822  \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}  \begin{funcdesc}{eigenvalues_and_eigenvectors}{a}
823  return the eigenvalues and eigenvectors of \var{a}. This is  return the eigenvalues and eigenvectors of \var{a}. This is
824  \begin{equation}  \begin{equation}
825  \code{matrixmult(a,V[:,i])=e[i]*V[:,i]}  \code{matrix_mult(a,V[:,i])=e[i]*V[:,i]}
826  \end{equation}  \end{equation}
827  where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are orthogonal and normalized, ie.  where \code{e,V=eigenvalues_and_eigenvectors(a)}. The eigenvectors \var{V} are orthogonal and normalized, ie.
828  \begin{equation}  \begin{equation}
829  \code{matrixmult(transpose(V),V)=kronecker(d)}  \code{matrix_mult(transpose(V),V)=kronecker(d)}
830  \end{equation}  \end{equation}
831  if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing size.  if \var{a} has shape \code{(d,d)}. The eigenvalues are ordered in increasing size.
832  The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.    The argument \var{a} has to be the symmetric, ie. \code{a=symmetric(a)}.  
# Line 867  and for a \RankFour arguments this is Line 867  and for a \RankFour arguments this is
867  \code{inner(a)}=\sum\hackscore{ijkl}\var{a0} \left[i,j,k,l\right]  \cdot \var{a1} \left[j,i,k,l\right]  \code{inner(a)}=\sum\hackscore{ijkl}\var{a0} \left[i,j,k,l\right]  \cdot \var{a1} \left[j,i,k,l\right]
868  \end{equation}  \end{equation}
869  \end{funcdesc}  \end{funcdesc}
870  \begin{funcdesc}{matrixmult}{a0,a1}  
871    \begin{funcdesc}{matrix_mult}{a0,a1}
872  returns the matrix product of \var{a0} and \var{a1}. If \var{a1} is \RankOne this is  returns the matrix product of \var{a0} and \var{a1}. If \var{a1} is \RankOne this is
873  \begin{equation}  \begin{equation}
874  \code{matrixmult(a)}\left[i\right]=\sum\hackscore{k}\var{a0}  \cdot \left[i,k\right]\var{a1} \left[k\right]  \code{matrix_mult(a)}\left[i\right]=\sum\hackscore{k}\var{a0}  \cdot \left[i,k\right]\var{a1} \left[k\right]
875    \end{equation}
876    and if \var{a1} is \RankTwo this is
877    \begin{equation}
878    \code{matrix_mult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0}  \cdot \left[i,k\right]\var{a1} \left[k,j\right]
879    \end{equation}
880    \end{funcdesc}
881    
882    \begin{funcdesc}{transposed_matrix_mult}{a0,a1}
883    returns the matrix product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
884    \code{matrix_mult(transpose(a0),a1)}.
885    If \var{a1} is \RankOne this is
886    \begin{equation}
887    \code{transposed_matrix_mult(a)}\left[i\right]=\sum\hackscore{k}\var{a0}  \cdot \left[k,i\right]\var{a1} \left[k\right]
888  \end{equation}  \end{equation}
889  and if \var{a1} is \RankTwo this is  and if \var{a1} is \RankTwo this is
890  \begin{equation}  \begin{equation}
891  \code{matrixmult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0}  \cdot \left[i,k\right]\var{a1} \left[k,j\right]  \code{transposed_matrix_mult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0}  \cdot \left[k,i\right]\var{a1} \left[k,j\right]
892  \end{equation}  \end{equation}
893  \end{funcdesc}  \end{funcdesc}
894    
895    \begin{funcdesc}{matrix_transposed_mult}{a0,a1}
896    returns the matrix product of \var{a0} and the transposed of \var{a1}.
897    The function is equivalent to
898    \code{matrix_mult(a0,transpose(a1))}.  
899    If \var{a1} is \RankTwo this is
900    \begin{equation}
901    \code{matrix_transposed_mult(a)}\left[i,j\right]=\sum\hackscore{k}\var{a0}  \cdot \left[i,k\right]\var{a1} \left[j,k\right]
902    \end{equation}
903    \end{funcdesc}
904    
905  \begin{funcdesc}{outer}{a0,a1}  \begin{funcdesc}{outer}{a0,a1}
906  returns the outer product of \var{a0} and \var{a1}. For instance if \var{a0} and \var{a1} both are \RankOne then  returns the outer product of \var{a0} and \var{a1}. For instance if \var{a0} and \var{a1} both are \RankOne then
907  \begin{equation}  \begin{equation}
# Line 887  and if \var{a0} is \RankOne and \var{a1} Line 912  and if \var{a0} is \RankOne and \var{a1}
912  \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]  \code{outer(a)}\left[i,j,k\right]=\var{a0} \left[i\right] \cdot \var{a1}\left[j,k\right]
913  \end{equation}  \end{equation}
914  \end{funcdesc}  \end{funcdesc}
915  \begin{funcdesc}{tensormult}{a0,a1}  
916    \begin{funcdesc}{tensor_mult}{a0,a1}
917  returns the tensor product of \var{a0} and \var{a1}. If \var{a1} is \RankTwo this is  returns the tensor product of \var{a0} and \var{a1}. If \var{a1} is \RankTwo this is
918  \begin{equation}  \begin{equation}
919  \code{tensormult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[k,l\right]  \code{tensor_mult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[k,l\right]
920  \end{equation}  \end{equation}
921  and if \var{a1} is \RankFour this is  and if \var{a1} is \RankFour this is
922  \begin{equation}  \begin{equation}
923  \code{tensormult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[m,n,k,l\right]  \code{tensor_mult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[m,n,k,l\right]
924  \end{equation}  \end{equation}
925  \end{funcdesc}  \end{funcdesc}
926    
927    \begin{funcdesc}{transposed_tensor_mult}{a0,a1}
928    returns the tensor product of the transposed of \var{a0} and \var{a1}. The function is equivalent to
929    \code{tensor_mult(transpose(a0),a1)}.
930    If \var{a1} is \RankTwo this is
931    \begin{equation}
932    \code{transposed_tensor_mult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[k,l,i,j\right] \cdot \var{a1} \left[k,l\right]
933    \end{equation}
934    and if \var{a1} is \RankFour this is
935    \begin{equation}
936    \code{transposed_tensor_mult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[m,n,i,j\right] \cdot \var{a1} \left[m,n,k,l\right]
937    \end{equation}
938    \end{funcdesc}
939    
940    \begin{funcdesc}{tensor_transposed_mult}{a0,a1}
941    returns the tensor product of \var{a0} and the transposed of \var{a1}.
942    The function is equivalent to
943    \code{tensor_mult(a0,transpose(a1))}.
944    If \var{a1} is \RankTwo this is
945    \begin{equation}
946    \code{tensor_transposed_mult(a)}\left[i,j\right]=\sum\hackscore{kl}\var{a0}\left[i,j,k,l\right] \cdot \var{a1} \left[l,k\right]
947    \end{equation}
948    and if \var{a1} is \RankFour this is
949    \begin{equation}
950    \code{tensor_transposed_mult(a)}\left[i,j,k,l\right]=\sum\hackscore{mn}\var{a0} \left[i,j,m,n\right] \cdot \var{a1} \left[k,l,m,n\right]
951    \end{equation}
952    \end{funcdesc}
953    
954  \begin{funcdesc}{grad}{a\optional{, where=None}}  \begin{funcdesc}{grad}{a\optional{, where=None}}
955  returns the gradient of \var{a}. If \var{where} is present the gradient will be calculated in \FunctionSpace \var{where} otherwise a  returns the gradient of \var{a}. If \var{where} is present the gradient will be calculated in \FunctionSpace \var{where} otherwise a
956  default \FunctionSpace is used. In case that \var{a} has \RankTwo one has  default \FunctionSpace is used. In case that \var{a} has \RankTwo one has
# Line 930  returns the divergence of \var{a}. This Line 984  returns the divergence of \var{a}. This
984  returns the jump of \var{a} over the discontinuity in its domain or if \Domain \var{domain} is present  returns the jump of \var{a} over the discontinuity in its domain or if \Domain \var{domain} is present
985  in \var{domain}.  in \var{domain}.
986  \begin{equation}  \begin{equation}
987  \code{jump(a)}=interpolate(a,FunctionOnContactOne(domain))-interpolate(a,FunctionOnContactZero(domain))  \begin{array}{rcl}
988    \code{jump(a)}& = &\code{interpolate(a,FunctionOnContactOne(domain))} \\
989                  &   & \hfill - \code{interpolate(a,FunctionOnContactZero(domain))}
990    \end{array}
991  \end{equation}  \end{equation}
992  \end{funcdesc}  \end{funcdesc}
993  \begin{funcdesc}{L2}{a}  \begin{funcdesc}{L2}{a}
994  returns the $L^2$-norm of \var{a} in its function space. This is  returns the $L^2$-norm of \var{a} in its function space. This is
995  \begin{equation}  \begin{equation}
996  \code{L2(a)}=integrate(length(a)^2) \; .  \code{L2(a)=integrate(length(a)}^2\code{)} \; .
997  \end{equation}  \end{equation}
998  \end{funcdesc}  \end{funcdesc}
999    
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