gbtrf#

sgbtrf

Functions

void sgbtrf(
    const INT           m,
    const INT           n,
    const INT           kl,
    const INT           ku,
          f32* restrict AB,
    const INT           ldab,
          INT* restrict ipiv,
          INT*          info
);

void sgbtrf(const INT m, const INT n, const INT kl, const INT ku, f32 *restrict AB, const INT ldab, INT *restrict ipiv, INT *info)#

SGBTRF computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges.

This is the blocked version of the algorithm, calling Level 3 BLAS.

The factorization has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

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

m

The number of rows of the matrix A. m>=0.

in

n

The number of columns 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.

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(m-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 min(m,n). The pivot indices; for 0<=i<min(m,n), row i of the matrix was interchanged with row ipiv[i].

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 division by zero will occur if it is used to solve a system of equations.

dgbtrf

Functions

void dgbtrf(
    const INT           m,
    const INT           n,
    const INT           kl,
    const INT           ku,
          f64* restrict AB,
    const INT           ldab,
          INT* restrict ipiv,
          INT*          info
);

void dgbtrf(const INT m, const INT n, const INT kl, const INT ku, f64 *restrict AB, const INT ldab, INT *restrict ipiv, INT *info)#

DGBTRF computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges.

This is the blocked version of the algorithm, calling Level 3 BLAS.

The factorization has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

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

m

The number of rows of the matrix A. m>=0.

in

n

The number of columns 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.

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(m-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 min(m,n). The pivot indices; for 0<=i<min(m,n), row i of the matrix was interchanged with row ipiv[i].

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 division by zero will occur if it is used to solve a system of equations.

cgbtrf

Functions

void cgbtrf(
    const INT           m,
    const INT           n,
    const INT           kl,
    const INT           ku,
          c64* restrict AB,
    const INT           ldab,
          INT* restrict ipiv,
          INT*          info
);

void cgbtrf(const INT m, const INT n, const INT kl, const INT ku, c64 *restrict AB, const INT ldab, INT *restrict ipiv, INT *info)#

CGBTRF computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges.

This is the blocked version of the algorithm, calling Level 3 BLAS.

The factorization has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

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

m

The number of rows of the matrix A. m>=0.

in

n

The number of columns 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.

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(m-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 min(m,n). The pivot indices; for 0<=i<min(m,n), row i of the matrix was interchanged with row ipiv[i].

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 division by zero will occur if it is used to solve a system of equations.

zgbtrf

Functions

void zgbtrf(
    const INT            m,
    const INT            n,
    const INT            kl,
    const INT            ku,
          c128* restrict AB,
    const INT            ldab,
          INT*  restrict ipiv,
          INT*           info
);

void zgbtrf(const INT m, const INT n, const INT kl, const INT ku, c128 *restrict AB, const INT ldab, INT *restrict ipiv, INT *info)#

ZGBTRF computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges.

This is the blocked version of the algorithm, calling Level 3 BLAS.

The factorization has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

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

m

The number of rows of the matrix A. m>=0.

in

n

The number of columns 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.

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(m-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 min(m,n). The pivot indices; for 0<=i<min(m,n), row i of the matrix was interchanged with row ipiv[i].

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 division by zero will occur if it is used to solve a system of equations.