gbsv#

sgbsv

Functions

void sgbsv(
    const INT           n,
    const INT           kl,
    const INT           ku,
    const INT           nrhs,
          f32* restrict AB,
    const INT           ldab,
          INT* restrict ipiv,
          f32* restrict B,
    const INT           ldb,
          INT*          info
);

void sgbsv(const INT n, const INT kl, const INT ku, const INT nrhs, f32 *restrict AB, const INT ldab, INT *restrict ipiv, f32 *restrict B, const INT ldb, INT *info)#

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

A * X = B

where A is a band matrix of order n with kl subdiagonals and ku superdiagonals, and X and B are n-by-nrhs matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as A = L * U, where L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. 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 m = n = 6, kl = 2, ku = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u03  u14  u25
    *    *    +    +    +    +       *    *   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
   a10  a21  a32  a43  a54   *      m10  m21  m32  m43  m54   *
   a20  a31  a42  a53   *    *      m20  m31  m42  m53   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges.

Parameters

in

n

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

in

kl

The number of subdiagonals within the band of A. kl>=0.

in

ku

The number of superdiagonals within the band of A. ku>=0.

in

nrhs

The number of right hand sides, i.e., the number of columns of the matrix B. nrhs>=0.

inout

AB

Array of dimension (ldab, n). On entry, the matrix A in band storage, in rows kl to 2*kl+ku; rows 0 to kl-1 of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB[kl+ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(n-1,j+kl). On exit, details of the factorization: U is stored as an upper triangular band matrix with kl+ku superdiagonals in rows 0 to kl+ku, and the multipliers used during the factorization are stored in rows kl+ku+1 to 2*kl+ku. See below for further details.

in

ldab

The leading dimension of the array AB. ldab>=2*kl+ku+1.

out

ipiv

Array of dimension n. The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row ipiv[i].

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, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed.

dgbsv

Functions

void dgbsv(
    const INT           n,
    const INT           kl,
    const INT           ku,
    const INT           nrhs,
          f64* restrict AB,
    const INT           ldab,
          INT* restrict ipiv,
          f64* restrict B,
    const INT           ldb,
          INT*          info
);

void dgbsv(const INT n, const INT kl, const INT ku, const INT nrhs, f64 *restrict AB, const INT ldab, INT *restrict ipiv, f64 *restrict B, const INT ldb, INT *info)#

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

A * X = B

where A is a band matrix of order n with kl subdiagonals and ku superdiagonals, and X and B are n-by-nrhs matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as A = L * U, where L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. 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 m = n = 6, kl = 2, ku = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u03  u14  u25
    *    *    +    +    +    +       *    *   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
   a10  a21  a32  a43  a54   *      m10  m21  m32  m43  m54   *
   a20  a31  a42  a53   *    *      m20  m31  m42  m53   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges.

Parameters

in

n

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

in

kl

The number of subdiagonals within the band of A. kl>=0.

in

ku

The number of superdiagonals within the band of A. ku>=0.

in

nrhs

The number of right hand sides, i.e., the number of columns of the matrix B. nrhs>=0.

inout

AB

Array of dimension (ldab, n). On entry, the matrix A in band storage, in rows kl to 2*kl+ku; rows 0 to kl-1 of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB[kl+ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(n-1,j+kl). On exit, details of the factorization: U is stored as an upper triangular band matrix with kl+ku superdiagonals in rows 0 to kl+ku, and the multipliers used during the factorization are stored in rows kl+ku+1 to 2*kl+ku. See below for further details.

in

ldab

The leading dimension of the array AB. ldab>=2*kl+ku+1.

out

ipiv

Array of dimension n. The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row ipiv[i].

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, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed.

cgbsv

Functions

void cgbsv(
    const INT           n,
    const INT           kl,
    const INT           ku,
    const INT           nrhs,
          c64* restrict AB,
    const INT           ldab,
          INT* restrict ipiv,
          c64* restrict B,
    const INT           ldb,
          INT*          info
);

void cgbsv(const INT n, const INT kl, const INT ku, const INT nrhs, c64 *restrict AB, const INT ldab, INT *restrict ipiv, c64 *restrict B, const INT ldb, INT *info)#

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

A * X = B

where A is a band matrix of order n with kl subdiagonals and ku superdiagonals, and X and B are n-by-nrhs matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as A = L * U, where L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. 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 m = n = 6, kl = 2, ku = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u03  u14  u25
    *    *    +    +    +    +       *    *   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
   a10  a21  a32  a43  a54   *      m10  m21  m32  m43  m54   *
   a20  a31  a42  a53   *    *      m20  m31  m42  m53   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges.

Parameters

in

n

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

in

kl

The number of subdiagonals within the band of A. kl>=0.

in

ku

The number of superdiagonals within the band of A. ku>=0.

in

nrhs

The number of right hand sides, i.e., the number of columns of the matrix B. nrhs>=0.

inout

AB

Array of dimension (ldab, n). On entry, the matrix A in band storage, in rows kl to 2*kl+ku; rows 0 to kl-1 of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB[kl+ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(n-1,j+kl). On exit, details of the factorization: U is stored as an upper triangular band matrix with kl+ku superdiagonals in rows 0 to kl+ku, and the multipliers used during the factorization are stored in rows kl+ku+1 to 2*kl+ku. See below for further details.

in

ldab

The leading dimension of the array AB. ldab>=2*kl+ku+1.

out

ipiv

Array of dimension n. The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row ipiv[i].

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, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed.

zgbsv

Functions

void zgbsv(
    const INT            n,
    const INT            kl,
    const INT            ku,
    const INT            nrhs,
          c128* restrict AB,
    const INT            ldab,
          INT*  restrict ipiv,
          c128* restrict B,
    const INT            ldb,
          INT*           info
);

void zgbsv(const INT n, const INT kl, const INT ku, const INT nrhs, c128 *restrict AB, const INT ldab, INT *restrict ipiv, c128 *restrict B, const INT ldb, INT *info)#

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

A * X = B

where A is a band matrix of order n with kl subdiagonals and ku superdiagonals, and X and B are n-by-nrhs matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as A = L * U, where L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. 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 m = n = 6, kl = 2, ku = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u03  u14  u25
    *    *    +    +    +    +       *    *   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
   a10  a21  a32  a43  a54   *      m10  m21  m32  m43  m54   *
   a20  a31  a42  a53   *    *      m20  m31  m42  m53   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges.

Parameters

in

n

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

in

kl

The number of subdiagonals within the band of A. kl>=0.

in

ku

The number of superdiagonals within the band of A. ku>=0.

in

nrhs

The number of right hand sides, i.e., the number of columns of the matrix B. nrhs>=0.

inout

AB

Array of dimension (ldab, n). On entry, the matrix A in band storage, in rows kl to 2*kl+ku; rows 0 to kl-1 of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB[kl+ku+i-j + j*ldab] = A(i,j) for max(0,j-ku)<=i<=min(n-1,j+kl). On exit, details of the factorization: U is stored as an upper triangular band matrix with kl+ku superdiagonals in rows 0 to kl+ku, and the multipliers used during the factorization are stored in rows kl+ku+1 to 2*kl+ku. See below for further details.

in

ldab

The leading dimension of the array AB. ldab>=2*kl+ku+1.

out

ipiv

Array of dimension n. The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row ipiv[i].

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, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed.