{S,D,C,Z}TBSV (uplo, trans, diag, n, k, a, lda, x, incx)
uplo character*1 On entry, specifies whether the matrix A is an upper- or lower-triangular matrix: If uplo = 'U' or 'u', A is an upper-triangular matrix. If uplo = 'L' or 'l', A is a lower-triangular matrix. On exit, uplo is unchanged. trans character*1 On entry, specifies the system to be solved: If trans = 'N' or 'n', the system is Ax = b. If trans = 'T' or 't', the system is transp(A)*x = b. If trans = 'C' or 'c', the system is conjug_transp(A)*x = b. On exit, trans is unchanged. diag character*1 On entry, specifies whether the matrix A is unit- triangular: If diag = 'U' or If diag = 'N' or 'n', A is not a unit-triangular matrix. On exit, diag is unchanged. n integer*4 On entry, the order of the matrix A; n >= 0. On exit, n is unchanged. k integer*4 On entry, if uplo is equal to 'U' or matrix A. If uplo is equal to 'L' or 'l', the number of sub-diagonals k of the matrix A; k >= 0. On exit, k is unchanged. a real*4 | real*8 | complex*8 | complex*16 On entry, a two-dimensional array with dimensions lda by n. When uplo specifies the upper portion of the matrix, the leading (k + 1) by n part of the array contains the upper-triangular band part of the matrix, supplied column by column. The main diagonal of the matrix is stored in row (k + 1) of the array, the first super-diagonal is stored in row k starting at position 2, and so on. The top left k by k triangle of the array A is not referenced. When uplo specifies the lower portion, the leading (k + 1) by n part of the array contains the lower-triangular band part of the matrix, supplied column by column. The main diagonal of the matrix is stored in row 1 of the array, the first sub-diagonal is stored in row 2 starting at position 1, and so on. The top right k by k triangle of the array A is not referenced. If diag is equal to 'U' or 'u', the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity. On exit, a is unchanged. lda integer*4 On entry, the first dimension of array A; lda >= (k+1). On exit, lda is unchanged. x real*4 | real*8 | complex*8 | complex*16 On entry, a one-dimensional array X of length at least (1+(n-1)*|incx|). Array X contains the vector b. On exit, x is overwritten with the solution vector x. incx integer*4 On entry, the increment for the elements of X; incx must not equal zero. On exit, incx is unchanged.
The _TBSV subprograms solve one of the following systems of linear equations for x: Ax = b or transp(A)*x = b . In addition to these operations, the CTBSV and ZTBSV subprograms solve the following system of linear equations for x: conjug_transp(A)*x = b. b and x are vectors with n elements and A is an n by n band matrix with (k + 1) diagonals. The matrix is a unit or non-unit, upper- or lower- triangular band matrix. The _TBSV routines do not perform checks for singularity or near singularity of the triangular matrix. The requirements for such a test depend on the application. If necessary, perform the test in your application program before calling the routine.
REAL*8 A(10,100), X(100) INCX = 1 K = 9 LDA = 10 N = 100 CALL DTBSV('L','T','U',N,K,A,LDA,X,INCX) This FORTRAN code solves the system transp(A)*x = b where A is a lower- triangular matrix, with a unit diagonal and 9 subdiagonals. The right hand side b is originally contained in the vector x.