cblas_?tbsv

Solves a system of linear equations whose coefficients are in a triangular band matrix.

Syntax

void cblas_stbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const float *a, const MKL_INT lda, float *x, const MKL_INT incx);

void cblas_dtbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const double *a, const MKL_INT lda, double *x, const MKL_INT incx);

void cblas_ctbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const void *a, const MKL_INT lda, void *x, const MKL_INT incx);

void cblas_ztbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const void *a, const MKL_INT lda, void *x, const MKL_INT incx);

Include Files

  • mkl.h

Description

The ?tbsv routines solve one of the following systems of equations:

A*x = b, or A'*x = b, or conjg(A')*x = b,

where:

b and x are n-element vectors,

A is an n-by-n unit, or non-unit, upper or lower triangular band matrix, with (k + 1) diagonals.

The routine does not test for singularity or near-singularity.

Such tests must be performed before calling this routine.

Input Parameters

Layout

Specifies whether two-dimensional array storage is row-major (CblasRowMajor) or column-major (CblasColMajor).

uplo

Specifies whether the matrix A is an upper or lower triangular matrix:

if uplo = CblasUpper the matrix is upper triangular;

if uplo = CblasLower, the matrix is low triangular.

trans

Specifies the system of equations:

if trans=CblasNoTrans, then A*x = b;

if trans=CblasTrans, then A'*x = b;

if trans=CblasConjTrans, then conjg(A')*x = b.

diag

Specifies whether the matrix A is unit triangular:

if diag = CblasUnit then the matrix is unit triangular;

if diag = CblasNonUnit, then the matrix is not unit triangular.

n

Specifies the order of the matrix A. The value of n must be at least zero.

k

On entry with uplo = CblasUpper, k specifies the number of super-diagonals of the matrix A. On entry with uplo = CblasLower, k specifies the number of sub-diagonals of the matrix A.

The value of k must satisfy 0k.

a

Array, size lda*n.

Layout = CblasColMajor:

Before entry with uplo = CblasUpper, the 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 of the array, the first super-diagonal starting at position 1 in row (k - 1), and so on. The top left k by k triangle of the array a is not referenced.

The following program segment transfers an upper triangular band matrix from conventional full matrix storage (matrix, with leading dimension ldm) to band storage (a, with leading dimension lda):

for (j = 0; j < n; j++) {
    m = k - j;
    for (i = max( 0, j - k); i <= j; i++) {
        a[(m+i) + j*lda] = matrix[i + j*ldm];
    }
}

Before entry with uplo = CblasLower, 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 0 of the array, the first sub-diagonal starting at position 0 in row 1, and so on. The bottom right k by k triangle of the array a is not referenced.

The following program segment transfers a lower triangular band matrix from conventional full matrix storage (matrix, with leading dimension ldm) to band storage (a, with leading dimension lda):

for (j = 0; j < n; j++) {
    m = -j;
    for (i = j; i < min(n, j + k + 1); i++) {
        a[(m+i) + j*lda] = matrix[i + j*ldm];
    }
}

When diag = CblasUnit, the elements of the array a corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.

Layout = CblasRowMajor:

Before entry with uplo = CblasUpper, the leading (k + 1)-by-n part of array a must contain the upper triangular band part of the matrix of coefficients. The matrix must be supplied row-by-row, with the leading diagonal of the matrix in column 0 of the array, the first super-diagonal starting at position 0 in column 1, and so on. The bottom right k-by-k triangle of array a is not referenced.

The following program segment transfers the upper triangular part of a Hermitian band matrix from row-major full matrix storage (matrix with leading dimension ldm) to row-major band storage (a, with leading dimension lda):

for (i = 0; i < n; i++) {
    m = -i;
    for (j = i; j < MIN(n, i+k+1); j++) {
        a[(m+j) + i*lda] = matrix[j + i*ldm];
    }
}

Before entry with uplo = CblasLower, the leading (k + 1)-by-n part of array a must contain the lower triangular band part of the matrix of coefficients, supplied row-by-row, with the leading diagonal of the matrix in column k of the array, the first sub-diagonal starting at position 1 in column k-1, and so on. The top left k-by-k triangle of array a is not referenced.

The following program segment transfers the lower triangular part of a Hermitian row-major band matrix from row-major full matrix storage (matrix, with leading dimension ldm) to row-major band storage (a, with leading dimension lda):

for (i = 0; i < n; i++) {
    m = k - i;
    for (j = max(0, i-k); j <= i; j++) {
         a[(m+j) + i*lda] = matrix[j + i*ldm];
    }
}
lda

Specifies the leading dimension of a as declared in the calling (sub)program. The value of lda must be at least (k + 1).

x

Array, size at least (1 + (n - 1)*abs(incx)). Before entry, the incremented array x must contain the n-element right-hand side vector b.

incx

Specifies the increment for the elements of x.

The value of incx must not be zero.

Output Parameters

x

Overwritten with the solution vector x.

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