sysv_rook

ssysv_rook

Functions

void ssysv_rook(
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          f32*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          f32*  restrict B,
    const INT            ldb,
          f32*  restrict work,
    const INT            lwork,
          INT*           info
);

void ssysv_rook(const char *uplo, const INT n, const INT nrhs, f32 *restrict A, const INT lda, INT *restrict ipiv, f32 *restrict B, const INT ldb, f32 *restrict work, const INT lwork, INT *info)

SSYSV_ROOK computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as A = U * D * U**T, if UPLO = ‘U’, or A = L * D * L**T, if UPLO = ‘L’, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

SSYTRF_ROOK is called to compute the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method.

The factored form of A is then used to solve the system of equations A * X = B by calling SSYTRS_ROOK.

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If uplo='L':

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1)

  is a 2-by-2 diagonal block.

  TRS will be done with Level 2 BLAS.

If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

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

nrhs

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

inout

A

Array of dimension (lda,n). On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if info=0, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by ssytrf_rook.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

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

work

Array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The length of work. lwork>=1, and for best performance lwork>=n*nb, where nb is the optimal block size for ssytrf_rook.

out

info

• info=0: successful exit

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

• info>0: if info=i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

dsysv_rook

Functions

void dsysv_rook(
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          f64*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          f64*  restrict B,
    const INT            ldb,
          f64*  restrict work,
    const INT            lwork,
          INT*           info
);

void dsysv_rook(const char *uplo, const INT n, const INT nrhs, f64 *restrict A, const INT lda, INT *restrict ipiv, f64 *restrict B, const INT ldb, f64 *restrict work, const INT lwork, INT *info)

DSYSV_ROOK computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as A = U * D * U**T, if UPLO = ‘U’, or A = L * D * L**T, if UPLO = ‘L’, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

DSYTRF_ROOK is called to compute the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method.

The factored form of A is then used to solve the system of equations A * X = B by calling DSYTRS_ROOK.

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If uplo='L':

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1)

  is a 2-by-2 diagonal block.

  TRS will be done with Level 2 BLAS.

If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

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

nrhs

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

inout

A

Array of dimension (lda,n). On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if info=0, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by dsytrf_rook.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

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

work

Array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The length of work. lwork>=1, and for best performance lwork>=n*nb, where nb is the optimal block size for dsytrf_rook.

out

info

• info=0: successful exit

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

• info>0: if info=i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

csysv_rook

Functions

void csysv_rook(
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          c64*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          c64*  restrict B,
    const INT            ldb,
          c64*  restrict work,
    const INT            lwork,
          INT*           info
);

void csysv_rook(const char *uplo, const INT n, const INT nrhs, c64 *restrict A, const INT lda, INT *restrict ipiv, c64 *restrict B, const INT ldb, c64 *restrict work, const INT lwork, INT *info)

CSYSV_ROOK computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as A = U * D * U**T, if UPLO = ‘U’, or A = L * D * L**T, if UPLO = ‘L’, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

CSYTRF_ROOK is called to compute the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method.

The factored form of A is then used to solve the system of equations A * X = B by calling CSYTRS_ROOK.

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If uplo='L':

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1)

  is a 2-by-2 diagonal block.

  TRS will be done with Level 2 BLAS.

If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

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

nrhs

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

inout

A

Complex array of dimension (lda,n). On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if info=0, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by csytrf_rook.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

inout

B

Complex 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

work

Complex array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The length of work. lwork>=1, and for best performance lwork>=n*nb, where nb is the optimal block size for csytrf_rook.

out

info

• info=0: successful exit

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

• info>0: if info=i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

zsysv_rook

Functions

void zsysv_rook(
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          c128* restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          c128* restrict B,
    const INT            ldb,
          c128* restrict work,
    const INT            lwork,
          INT*           info
);

void zsysv_rook(const char *uplo, const INT n, const INT nrhs, c128 *restrict A, const INT lda, INT *restrict ipiv, c128 *restrict B, const INT ldb, c128 *restrict work, const INT lwork, INT *info)

ZSYSV_ROOK computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as A = U * D * U**T, if UPLO = ‘U’, or A = L * D * L**T, if UPLO = ‘L’, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

ZSYTRF_ROOK is called to compute the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method.

The factored form of A is then used to solve the system of equations A * X = B by calling ZSYTRS_ROOK.

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If uplo='L':

• If ipiv[k]>=0, then rows and columns k and ipiv[k] were interchanged and D(k,k) is a 1-by-1 diagonal block.

• If ipiv[k]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1)

  is a 2-by-2 diagonal block.

  TRS will be done with Level 2 BLAS.

If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

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

nrhs

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

inout

A

Complex array of dimension (lda,n). On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if info=0, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by zsytrf_rook.

in

lda

The leading dimension of the array A. lda>=max(1,n).

out

ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

inout

B

Complex 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

work

Complex array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The length of work. lwork>=1, and for best performance lwork>=n*nb, where nb is the optimal block size for zsytrf_rook.

out

info

• info=0: successful exit

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

• info>0: if info=i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.