_TBMV,_TBSV(3F) _TBMV,_TBSV(3F)NAME
dtbmv, stbmv, ztbmv, ctbmv, dtbsv, stbsv, ztbsv, ctbsv - BLAS Level Two
Matrix-Vector Product and Solution of System of Equations.
FORTRAN 77 SYNOPSIS
subroutine dtbmv( uplo, trans, diag, n, k, a, lda, x, incx )
subroutine dtbsv( uplo, trans, diag, n, k, a, lda, x, incx )
character*1 uplo, trans, diag
integer n, k, lda, incx
double precision a( lda,*), x(*)
subroutine stbmv( uplo, trans, diag, n, k, a, lda, x, incx )
subroutine stbsv( uplo, trans, diag, n, k, a, lda, x, incx )
character*1 uplo, trans, diag
integer n, k, lda, incx
real a( lda,*), x(*)
subroutine ztbmv( uplo, trans, diag, n, k, a, lda, x, incx )
subroutine ztbsv( uplo, trans, diag, n, k, a, lda, x, incx )
character*1 uplo, trans, diag
integer n, k, lda, incx
double complex a( lda,*), x(*)
subroutine ctbmv( uplo, trans, diag, n, a, k, lda, x, incx )
subroutine ctbsv( uplo, trans, diag, n, a, k, lda, x, incx )
character*1 uplo, trans, diag
integer n, k, lda, incx
complex a( lda,*), x(*)
C SYNOPSIS
void dtbmv( uplo, trans, diag, n, k, a, lda, x, incx )
void dtbsv( uplo, trans, diag, n, k, a, lda, x, incx )
MatrixTriangle uplo;
MatrixTranspose trans;
MatrixUnitTriangular diag;
Integer n, k, lda, incx;
double (*a)[lda*k], (*x)[ n ];
void stbmv( uplo, trans, diag, n, k, a, lda, x, incx )
void stbsv( uplo, trans, diag, n, k, a, lda, x, incx )
MatrixTriangle uplo;
MatrixTranspose trans;
MatrixUnitTriangular diag;
Integer n, k, lda, incx;
float (*a)[lda*k], (*x)[ n ];
void ztbmv( uplo, trans, diag, n, k, a, lda, x, incx )
void ztbsv( uplo, trans, diag, n, k, a, lda, x, incx )
MatrixTriangle uplo;
MatrixTranspose trans;
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MatrixUnitTriangular diag;
Integer n, k, lda, incx;
Zomplex (*a)[lda*k], (*x)[ n ];
void ctbmv( uplo, trans, diag, n, k, a, lda, x, incx )
void ctbsv( uplo, trans, diag, n, k, a, lda, x, incx )
MatrixTriangle uplo;
MatrixTranspose trans;
MatrixUnitTriangular diag;
Integer n, k, lda, incx;
Complex (*a)[lda*k], (*x)[ n ];
DESCRIPTION
dtbmv, stbmv, ztbmv and ctbmv perform one of the matrix-vector operations
x := A*x, or x := A'*x, or x := conjg( A' )*x,
where x is an n element vector and A is an n by n unit, or non-unit,
upper or lower triangular band matrix, with ( k + 1 ) diagonals.
dtbsv, stbsv, ztbsv and ctbsv solve one of the systems of equations
A*x = b, or A'*x = b, or conjg( A' )*x = b,
where b and x are n element vectors and A is an n by n unit, or non-unit,
upper or lower triangular band matrix, with ( k + 1 ) diagonals. No test
for singularity or near-singularity is included in these routines. Such
tests must be performed before calling these routines.
PARAMETERS
uplo On entry, uplo specifies whether the matrix is an upper or lower
triangular matrix as follows:
FORTRAN
uplo = 'U' or 'u' A is an upper triangular matrix.
uplo = 'L' or 'l' A is a lower triangular matrix.
C
uplo = UpperTriangle A is an upper triangular matrix.
uplo = LowerTriangle A is a lower triangular matrix.
Unchanged on exit.
trans On entry, trans specifies the operation to be
FORTRAN
trans = 'N' or 'n' x := A*x / A*x = b.
trans = 'T' or 't' x := A'*x / A'*x = b.
trans = 'C' or 'c' x := conjg( A' )*x /
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conjg( A' )*x = b.
C
trans = NoTranspose x := A*x / A*x = b.
trans = Transpose x := A'*x / A'*x = b.
trans = ConjugateTranspose x := conjg( A' )*x /
conjg( A' )*x = b.
For real value matrices, trans='C' and trans='T' has the same
meaning.
Unchanged on exit.
diag On entry, diag specifies whether or not A is unit triangular as
follows:
FORTRAN
diag = 'U' or 'u' A is assumed to be unit triangular.
diag = 'N' or 'n' A is not assumed to be unit triangular.
C
diag = UnitTriangular A is assumed to be unit
triangular.
diag = NotUnitTriangular A is not assumed to be unit
triangular.
Unchanged on exit.
n On entry, n specifies the order of the matrix A. n must be at
least zero.
Unchanged on exit.
k On entry, with uplo = 'U' or 'u' or UpperTriangle , k specifies
the number of super-diagonals of the matrix A.
On entry with uplo = 'L' or 'l' or LowerTriangle , k specifies
the number of sub-diagonals of the matrix A.
k must satisfy 0 .le. k.
Unchanged on exit.
a An array containing the matrix A.
FORTRAN
Array of dimension ( lda, n ).
C
A pointer to an array of size lda*n containing the matrix A.
See note below about array storage convention for C.
Before entry with uplo = 'U' or 'u' or UpperTriangle , the
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leading ( k + 1 ) by n part of the array a must contain the upper
triangular band part of the matrix of coefficients, supplied
column by column, with the leading diagonal of the matrix in row
( k + 1 ) of the array, the first super-diagonal starting at
position 2 in row k, and so on. The top left k by k triangle of
the array a is not referenced. The following Fortran program
segment will transfer an upper triangular band matrix from
conventional full matrix storage to band storage:
DO 20, J = 1, N
M = K + 1 - J
DO 10, I = MAX( 1, J - K ), J
A( M + I, J ) = matrix( I, J )
10 CONTINUE
20 CONTINUE
Before entry with uplo = 'L' or 'l'or LowerTraingle , the leading
( k + 1 ) by n part of the array a must contain the lower
triangular band part of the matrix of coefficients, supplied
column by column, with the leading diagonal of the matrix in row
1 of the array, the first sub-diagonal starting at position 1 in
row 2, and so on. The bottom right k by k triangle of the array a
is not referenced. The following Fortran program segment will
transfer a lower triangular band matrix from conventional full
matrix storage to band storage:
DO 20, J = 1, N
M = 1 - J
DO 10, I = J, MIN( N, J + K )
A( M + I, J ) = matrix( I, J )
10 CONTINUE
20 CONTINUE
Note that when diag = 'U' or 'u' or , the elements of a
corresponding to the diagonal elements of the matrix A are not
referenced either, but are assumed to be unity.
Unchanged on exit.
lda On entry, lda specifies the first dimension of A as declared in
the calling (sub) program. lda must be at least max( 1, n ).
Unchanged on exit.
x Array of size at least ( 1 + ( n - 1 )*abs( incx ) ). Before
entry, the incremented array x must contain the vector x. On
exit, x is overwritten with the transformed/solution vector x.
incx On entry, incx specifies the increment for the elements of x.
incx must not be zero.
Unchanged on exit.
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_TBMV,_TBSV(3F) _TBMV,_TBSV(3F)C ARRAY STORAGE CONVENTION
The matrices are assumed to be stored in a one dimensional C array
in an analogous fashion as a Fortran array (column major). Therefore,
the element A(i+1,j) of matrix A is stored immediately after the
element A(i,j), while A(i,j+1) is lda elements apart from A(i,j).
The element A(i,j) of the matrix can be accessed directly by reference
to a[ (j-1)*lda + (i-1) ].
AUTHORS
Jack Dongarra, Argonne National Laboratory.
Iain Duff, AERE Harwell.
Jeremy Du Croz, Numerical Algorithms Group Ltd.
Sven Hammarling, Numerical Algorithms Group Ltd.
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