Solve.banded {limSolve} | R Documentation |
Solves the linear system of equations
Ax=B
by Gaussion elimination
where A has to be square, and banded, i.e. with the only nonzero elements in bands near the diagonal.
The matrix A is either inputted as a full square matrix or as the non-zero bands
uses subroutines dgbfa and dgbsl (FORTRAN) from the Linpack routines
Solve.banded(abd, nup, nlow,rhs=rep(0,times=ncol(abd)), full=(nrow(abd)==ncol(abd)))
abd |
either a matrix containing the (nonzero) bands, rotated row-wise (anti-clockwise) only or a full square matrix |
nup |
number of nonzero bands above the diagonal; ignored if full matrix is inputted |
nlow |
number of nonzero bands below the diagonal; ignored if full matrix is inputted |
rhs |
numeric vector containing right hand side |
full |
if true: full matrix is passed, if false:banded |
If the input matrix abd
is square, it is assumed that the full, square A is inputted, unless full
is false.
is abd
is not square, then the number of columns denote the number of unknowns, while the number of rows equals the nonzero bands, i.e. nup+nlow+1
vector with solution, x, of banded system of equations Ax=B
Karline Soetaert <k.soetaert@nioo.knaw.nl>
J.J. Dongarra, J.R. Bunch, C.B. Moler, G.W. Stewart, LINPACK Users' Guide, SIAM, 1979.
Solve.tridiag
To solve a tridiagonal system of linear equations.
Solve
the generalised inverse solution,
solve
the R default
# Generate a banded matrix of random numbers nup <- 2 # nr nonzero bands above diagonal ndwn <- 3 # nr nonzero bands below diagonal nn <- 10 # nr rows and columns of A A <- matrix(nrow=nn,ncol=nn,data=runif(1:(nn*nn))) A [row(A)<col(A)-nup | row(A)>col(A)+ndwn]<-0 diag(A) <- 1 # 1 on diagonal is easily recognised # right hand side B<-runif(nrow(A)) # solve it, using the default solver and banded (inputting full matrix) Full <- solve(A,B) Band1<- Solve.banded(A,nup,ndwn,B) # create banded form of matrix A Aext <- rbind(matrix(nc=ncol(A),nr=nup,0), A,matrix(nc=ncol(A),nr=ndwn,0)) abd <- matrix(nrow=nup+ndwn+1,ncol=nn, data=Aext[(col(Aext))<=row(Aext)&col(Aext)>=row(Aext)-ndwn-nup]) # print both to screen A abd # solve problem with banded version Band2 <- Solve.banded(abd,nup,ndwn,B) # compare 3 methods of solution rbind(Full,Band1,Band2)