# Solution of a banded system of linear equations Solves the linear system of equations [REMOVE_ME]$$ Ax = B [REMOVE_ME_2]$$ 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 lapack subroutine dgbsv (FORTRAN) ## Description 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 lapack subroutine dgbsv (FORTRAN) ```r Solve.banded(abd, nup, nlow, B = rep(0, times = ncol(abd)), full = (nrow(abd) == ncol(abd))) ``` ## Arguments - `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. - `B`: Right-hand side of the equations, a vector with length = number of rows of `A`, or a matrix with number of rows = number of rows of `A`. - `full`: if `TRUE`: full matrix is inputted, if `FALSE`: banded matrix is input. ## Returns matrix with the solution, `X`, of the banded system of equations A X =B, the number of columns of this matrix = number of columns of `B`. ## Author(s) Karline Soetaert ## Note A similar function but that requires a totally different input can now also be found in the `Matrix` package ## Examples ```r # 1. Generate a banded matrix of random numbers, full format 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) # 2. create banded form of matrix A Aext <- rbind(matrix(ncol = ncol(A), nrow = nup, 0), A, matrix(ncol = ncol(A), nrow = 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 cbind(Full, Band1, Band2) # same, now with 3 different right hand sides B3 <- cbind(B, B*2, B*3) Solve.banded(abd, nup, ndwn, B3) ``` ## References J.J. Dongarra, J.R. Bunch, C.B. Moler, G.W. Stewart, LINPACK Users' Guide, SIAM, 1979. ## Details If the input matrix `abd` is square, it is assumed that the full, square A is inputted, unless `full` is set to `FALSE`. If `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 ## See Also `Solve.tridiag` to solve a tridiagonal system of linear equations. `Solve` the generalised inverse solution, `solve` the R default

Solve.banded function