pbsv#

Functions

void spbsv(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          f32*  restrict AB,
    const INT            ldab,
          f32*  restrict B,
    const INT            ldb,
          INT*           info
);
void spbsv(const char *uplo, const INT n, const INT kd, const INT nrhs, f32 *restrict AB, const INT ldab, f32 *restrict B, const INT ldb, INT *info)#

SPBSV computes the solution to a real system of linear equations. 

A * X = B

where A is an n-by-n symmetric positive definite band matrix and X and B are n-by-nrhs matrices.

The Cholesky decomposition is used to factor A as 

A = U**T * U,  if uplo = 'U', or
A = L * L**T,  if uplo = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

Further Details:

The band storage scheme is illustrated by the following example, when n=6, kd=2, and uplo='U':

On entry:                       On exit:

     *    *   a02  a13  a24  a35      *    *   u02  u13  u24  u35
     *   a01  a12  a23  a34  a45      *   u01  u12  u23  u34  u45
    a00  a11  a22  a33  a44  a55     u00  u11  u22  u33  u44  u55

Similarly, if uplo='L' the format of A is as follows:

On entry:                       On exit:

    a00  a11  a22  a33  a44  a55     l00  l11  l22  l33  l44  l55
    a10  a21  a32  a43  a54   *      l10  l21  l32  l43  l54   *
    a20  a31  a42  a53   *    *      l20  l31  l42  l53   *    *

Array elements marked * are not used by the routine.

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in
kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in
nrhs

The number of right hand sides. nrhs>=0.

inout
AB

Array of dimension (ldab, n). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if info=0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, in the same storage format as A.

in
ldab

The leading dimension of the array AB. ldab>=kd+1.

inout
B

Array of dimension (ldb, nrhs). On entry, the n-by-nrhs right hand side matrix B. On exit, if info=0, the n-by-nrhs solution matrix X.

in
ldb

The leading dimension of the array B. ldb>=max(1,n).

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed.

Functions

void dpbsv(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          f64*  restrict AB,
    const INT            ldab,
          f64*  restrict B,
    const INT            ldb,
          INT*           info
);
void dpbsv(const char *uplo, const INT n, const INT kd, const INT nrhs, f64 *restrict AB, const INT ldab, f64 *restrict B, const INT ldb, INT *info)#

DPBSV computes the solution to a real system of linear equations. 

A * X = B

where A is an n-by-n symmetric positive definite band matrix and X and B are n-by-nrhs matrices.

The Cholesky decomposition is used to factor A as 

A = U**T * U,  if uplo = 'U', or
A = L * L**T,  if uplo = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

Further Details:

The band storage scheme is illustrated by the following example, when n=6, kd=2, and uplo='U':

On entry:                       On exit:

     *    *   a02  a13  a24  a35      *    *   u02  u13  u24  u35
     *   a01  a12  a23  a34  a45      *   u01  u12  u23  u34  u45
    a00  a11  a22  a33  a44  a55     u00  u11  u22  u33  u44  u55

Similarly, if uplo='L' the format of A is as follows:

On entry:                       On exit:

    a00  a11  a22  a33  a44  a55     l00  l11  l22  l33  l44  l55
    a10  a21  a32  a43  a54   *      l10  l21  l32  l43  l54   *
    a20  a31  a42  a53   *    *      l20  l31  l42  l53   *    *

Array elements marked * are not used by the routine.

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in
kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in
nrhs

The number of right hand sides. nrhs>=0.

inout
AB

Array of dimension (ldab, n). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if info=0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, in the same storage format as A.

in
ldab

The leading dimension of the array AB. ldab>=kd+1.

inout
B

Array of dimension (ldb, nrhs). On entry, the n-by-nrhs right hand side matrix B. On exit, if info=0, the n-by-nrhs solution matrix X.

in
ldb

The leading dimension of the array B. ldb>=max(1,n).

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed.

Functions

void cpbsv(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          c64*  restrict AB,
    const INT            ldab,
          c64*  restrict B,
    const INT            ldb,
          INT*           info
);
void cpbsv(const char *uplo, const INT n, const INT kd, const INT nrhs, c64 *restrict AB, const INT ldab, c64 *restrict B, const INT ldb, INT *info)#

CPBSV computes the solution to a complex system of linear equations. 

A * X = B

where A is an n-by-n Hermitian positive definite band matrix and X and B are n-by-nrhs matrices.

The Cholesky decomposition is used to factor A as 

A = U**H * U,  if uplo = 'U', or
A = L * L**H,  if uplo = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

Further Details:

The band storage scheme is illustrated by the following example, when n=6, kd=2, and uplo='U':

On entry:                       On exit:

     *    *   a02  a13  a24  a35      *    *   u02  u13  u24  u35
     *   a01  a12  a23  a34  a45      *   u01  u12  u23  u34  u45
    a00  a11  a22  a33  a44  a55     u00  u11  u22  u33  u44  u55

Similarly, if uplo='L' the format of A is as follows:

On entry:                       On exit:

    a00  a11  a22  a33  a44  a55     l00  l11  l22  l33  l44  l55
    a10  a21  a32  a43  a54   *      l10  l21  l32  l43  l54   *
    a20  a31  a42  a53   *    *      l20  l31  l42  l53   *    *

Array elements marked * are not used by the routine.

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in
kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in
nrhs

The number of right hand sides. nrhs>=0.

inout
AB

Array of dimension (ldab, n). On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if info=0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A.

in
ldab

The leading dimension of the array AB. ldab>=kd+1.

inout
B

Array of dimension (ldb, nrhs). On entry, the n-by-nrhs right hand side matrix B. On exit, if info=0, the n-by-nrhs solution matrix X.

in
ldb

The leading dimension of the array B. ldb>=max(1,n).

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed.

Functions

void zpbsv(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          c128* restrict AB,
    const INT            ldab,
          c128* restrict B,
    const INT            ldb,
          INT*           info
);
void zpbsv(const char *uplo, const INT n, const INT kd, const INT nrhs, c128 *restrict AB, const INT ldab, c128 *restrict B, const INT ldb, INT *info)#

ZPBSV computes the solution to a complex system of linear equations. 

A * X = B

where A is an n-by-n Hermitian positive definite band matrix and X and B are n-by-nrhs matrices.

The Cholesky decomposition is used to factor A as 

A = U**H * U,  if uplo = 'U', or
A = L * L**H,  if uplo = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

Further Details:

The band storage scheme is illustrated by the following example, when n=6, kd=2, and uplo='U':

On entry:                       On exit:

     *    *   a02  a13  a24  a35      *    *   u02  u13  u24  u35
     *   a01  a12  a23  a34  a45      *   u01  u12  u23  u34  u45
    a00  a11  a22  a33  a44  a55     u00  u11  u22  u33  u44  u55

Similarly, if uplo='L' the format of A is as follows:

On entry:                       On exit:

    a00  a11  a22  a33  a44  a55     l00  l11  l22  l33  l44  l55
    a10  a21  a32  a43  a54   *      l10  l21  l32  l43  l54   *
    a20  a31  a42  a53   *    *      l20  l31  l42  l53   *    *

Array elements marked * are not used by the routine.

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in
kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in
nrhs

The number of right hand sides. nrhs>=0.

inout
AB

Array of dimension (ldab, n). On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if info=0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A.

in
ldab

The leading dimension of the array AB. ldab>=kd+1.

inout
B

Array of dimension (ldb, nrhs). On entry, the n-by-nrhs right hand side matrix B. On exit, if info=0, the n-by-nrhs solution matrix X.

in
ldb

The leading dimension of the array B. ldb>=max(1,n).

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed.