sysvx

ssysvx

Functions

void ssysvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const f32*  restrict A,
    const INT            lda,
          f32*  restrict AF,
    const INT            ldaf,
          INT*  restrict ipiv,
    const f32*  restrict B,
    const INT            ldb,
          f32*  restrict X,
    const INT            ldx,
          f32*           rcond,
          f32*  restrict ferr,
          f32*  restrict berr,
          f32*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
          INT*           info
);

void ssysvx(const char *fact, const char *uplo, const INT n, const INT nrhs, const f32 *restrict A, const INT lda, f32 *restrict AF, const INT ldaf, INT *restrict ipiv, const f32 *restrict B, const INT ldb, f32 *restrict X, const INT ldx, f32 *rcond, f32 *restrict ferr, f32 *restrict berr, f32 *restrict work, const INT lwork, INT *restrict iwork, INT *info)

SSYSVX uses the diagonal pivoting factorization to compute 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.

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed:

1. If FACT = ‘N’, the diagonal pivoting method is used to factor A. The form of the factorization is 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.

2. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form of A.

4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Parameters

in

fact

Specifies whether the factored form of A has been supplied on entry. = ‘F’: On entry, AF and IPIV contain the factored form. = ‘N’: The matrix A will be copied to AF and factored.

in

uplo

= ‘U’: Upper triangle of A is stored = ‘L’: Lower triangle of A is stored

in

n

The number of linear equations. n >= 0.

in

nrhs

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

in

A

The symmetric matrix A. Double precision array, dimension (lda, n).

in

lda

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

inout

AF

If fact = ‘F’, contains the factored form from SSYTRF. If fact = ‘N’, on exit contains the factored form. Double precision array, dimension (ldaf, n).

in

ldaf

The leading dimension of the array AF. ldaf >= max(1, n).

inout

ipiv

If fact = ‘F’, contains the pivot indices from SSYTRF. If fact = ‘N’, on exit contains the pivot indices. Integer array, dimension (n).

in

B

The right hand side matrix B. Double precision array, dimension (ldb, nrhs).

in

ldb

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

out

X

The solution matrix X. Double precision array, dimension (ldx, nrhs).

in

ldx

The leading dimension of the array X. ldx >= max(1, n).

out

rcond

The reciprocal condition number estimate of A.

out

ferr

Forward error bound for each solution vector. Double precision array, dimension (nrhs).

out

berr

Backward error for each solution vector. Double precision array, dimension (nrhs).

out

work

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

in

lwork

The length of work. lwork >= max(1, 3*n), and for best performance lwork >= max(1, 3*n, n*nb). If lwork = -1, a workspace query is assumed.

out

iwork

Integer array, dimension (n).

out

info

• = 0: successful exit

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

• > 0: if info = i, and i is

• <= N: D(i,i) is exactly zero. The factorization has been completed but D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.

• = N+1: D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision.

dsysvx

Functions

void dsysvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const f64*  restrict A,
    const INT            lda,
          f64*  restrict AF,
    const INT            ldaf,
          INT*  restrict ipiv,
    const f64*  restrict B,
    const INT            ldb,
          f64*  restrict X,
    const INT            ldx,
          f64*           rcond,
          f64*  restrict ferr,
          f64*  restrict berr,
          f64*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
          INT*           info
);

void dsysvx(const char *fact, const char *uplo, const INT n, const INT nrhs, const f64 *restrict A, const INT lda, f64 *restrict AF, const INT ldaf, INT *restrict ipiv, const f64 *restrict B, const INT ldb, f64 *restrict X, const INT ldx, f64 *rcond, f64 *restrict ferr, f64 *restrict berr, f64 *restrict work, const INT lwork, INT *restrict iwork, INT *info)

DSYSVX uses the diagonal pivoting factorization to compute 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.

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed:

1. If FACT = ‘N’, the diagonal pivoting method is used to factor A. The form of the factorization is 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.

2. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form of A.

4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Parameters

in

fact

Specifies whether the factored form of A has been supplied on entry. = ‘F’: On entry, AF and IPIV contain the factored form. = ‘N’: The matrix A will be copied to AF and factored.

in

uplo

= ‘U’: Upper triangle of A is stored = ‘L’: Lower triangle of A is stored

in

n

The number of linear equations. n >= 0.

in

nrhs

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

in

A

The symmetric matrix A. Double precision array, dimension (lda, n).

in

lda

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

inout

AF

If fact = ‘F’, contains the factored form from DSYTRF. If fact = ‘N’, on exit contains the factored form. Double precision array, dimension (ldaf, n).

in

ldaf

The leading dimension of the array AF. ldaf >= max(1, n).

inout

ipiv

If fact = ‘F’, contains the pivot indices from DSYTRF. If fact = ‘N’, on exit contains the pivot indices. Integer array, dimension (n).

in

B

The right hand side matrix B. Double precision array, dimension (ldb, nrhs).

in

ldb

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

out

X

The solution matrix X. Double precision array, dimension (ldx, nrhs).

in

ldx

The leading dimension of the array X. ldx >= max(1, n).

out

rcond

The reciprocal condition number estimate of A.

out

ferr

Forward error bound for each solution vector. Double precision array, dimension (nrhs).

out

berr

Backward error for each solution vector. Double precision array, dimension (nrhs).

out

work

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

in

lwork

The length of work. lwork >= max(1, 3*n), and for best performance lwork >= max(1, 3*n, n*nb). If lwork = -1, a workspace query is assumed.

out

iwork

Integer array, dimension (n).

out

info

• = 0: successful exit

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

• > 0: if info = i, and i is

• <= N: D(i,i) is exactly zero. The factorization has been completed but D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.

• = N+1: D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision.

csysvx

Functions

void csysvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const c64*  restrict A,
    const INT            lda,
          c64*  restrict AF,
    const INT            ldaf,
          INT*  restrict ipiv,
    const c64*  restrict B,
    const INT            ldb,
          c64*  restrict X,
    const INT            ldx,
          f32*           rcond,
          f32*  restrict ferr,
          f32*  restrict berr,
          c64*  restrict work,
    const INT            lwork,
          f32*  restrict rwork,
          INT*           info
);

void csysvx(const char *fact, const char *uplo, const INT n, const INT nrhs, const c64 *restrict A, const INT lda, c64 *restrict AF, const INT ldaf, INT *restrict ipiv, const c64 *restrict B, const INT ldb, c64 *restrict X, const INT ldx, f32 *rcond, f32 *restrict ferr, f32 *restrict berr, c64 *restrict work, const INT lwork, f32 *restrict rwork, INT *info)

CSYSVX uses the diagonal pivoting factorization to compute 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.

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed:

1. If FACT = ‘N’, the diagonal pivoting method is used to factor A. The form of the factorization is 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.

2. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form of A.

4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Parameters

in

fact

Specifies whether the factored form of A has been supplied on entry. = ‘F’: On entry, AF and IPIV contain the factored form. = ‘N’: The matrix A will be copied to AF and factored.

in

uplo

= ‘U’: Upper triangle of A is stored = ‘L’: Lower triangle of A is stored

in

n

The number of linear equations. n >= 0.

in

nrhs

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

in

A

The symmetric matrix A. Complex*16 array, dimension (lda, n).

in

lda

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

inout

AF

If fact = ‘F’, contains the factored form from CSYTRF. If fact = ‘N’, on exit contains the factored form. Complex*16 array, dimension (ldaf, n).

in

ldaf

The leading dimension of the array AF. ldaf >= max(1, n).

inout

ipiv

If fact = ‘F’, contains the pivot indices from CSYTRF. If fact = ‘N’, on exit contains the pivot indices. Integer array, dimension (n).

in

B

The right hand side matrix B. Complex*16 array, dimension (ldb, nrhs).

in

ldb

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

out

X

The solution matrix X. Complex*16 array, dimension (ldx, nrhs).

in

ldx

The leading dimension of the array X. ldx >= max(1, n).

out

rcond

The reciprocal condition number estimate of A.

out

ferr

Forward error bound for each solution vector. Single precision array, dimension (nrhs).

out

berr

Backward error for each solution vector. Single precision array, dimension (nrhs).

out

work

Complex*16 array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns the optimal lwork.

in

lwork

The length of work. lwork >= max(1, 2*n), and for best performance lwork >= max(1, 2*n, n*nb). If lwork = -1, a workspace query is assumed.

out

rwork

Single precision array, dimension (n).

out

info

• = 0: successful exit

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

• > 0: if info = i, and i is

• <= N: D(i,i) is exactly zero. The factorization has been completed but D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.

• = N+1: D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision.

zsysvx

Functions

void zsysvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
    const c128* restrict A,
    const INT            lda,
          c128* restrict AF,
    const INT            ldaf,
          INT*  restrict ipiv,
    const c128* restrict B,
    const INT            ldb,
          c128* restrict X,
    const INT            ldx,
          f64*           rcond,
          f64*  restrict ferr,
          f64*  restrict berr,
          c128* restrict work,
    const INT            lwork,
          f64*  restrict rwork,
          INT*           info
);

void zsysvx(const char *fact, const char *uplo, const INT n, const INT nrhs, const c128 *restrict A, const INT lda, c128 *restrict AF, const INT ldaf, INT *restrict ipiv, const c128 *restrict B, const INT ldb, c128 *restrict X, const INT ldx, f64 *rcond, f64 *restrict ferr, f64 *restrict berr, c128 *restrict work, const INT lwork, f64 *restrict rwork, INT *info)

ZSYSVX uses the diagonal pivoting factorization to compute 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.

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed:

1. If FACT = ‘N’, the diagonal pivoting method is used to factor A. The form of the factorization is 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.

2. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form of A.

4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Parameters

in

fact

Specifies whether the factored form of A has been supplied on entry. = ‘F’: On entry, AF and IPIV contain the factored form. = ‘N’: The matrix A will be copied to AF and factored.

in

uplo

= ‘U’: Upper triangle of A is stored = ‘L’: Lower triangle of A is stored

in

n

The number of linear equations. n >= 0.

in

nrhs

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

in

A

The symmetric matrix A. Complex*16 array, dimension (lda, n).

in

lda

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

inout

AF

If fact = ‘F’, contains the factored form from ZSYTRF. If fact = ‘N’, on exit contains the factored form. Complex*16 array, dimension (ldaf, n).

in

ldaf

The leading dimension of the array AF. ldaf >= max(1, n).

inout

ipiv

If fact = ‘F’, contains the pivot indices from ZSYTRF. If fact = ‘N’, on exit contains the pivot indices. Integer array, dimension (n).

in

B

The right hand side matrix B. Complex*16 array, dimension (ldb, nrhs).

in

ldb

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

out

X

The solution matrix X. Complex*16 array, dimension (ldx, nrhs).

in

ldx

The leading dimension of the array X. ldx >= max(1, n).

out

rcond

The reciprocal condition number estimate of A.

out

ferr

Forward error bound for each solution vector. Double precision array, dimension (nrhs).

out

berr

Backward error for each solution vector. Double precision array, dimension (nrhs).

out

work

Complex*16 array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns the optimal lwork.

in

lwork

The length of work. lwork >= max(1, 2*n), and for best performance lwork >= max(1, 2*n, n*nb). If lwork = -1, a workspace query is assumed.

out

rwork

Double precision array, dimension (n).

out

info

• = 0: successful exit

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

• > 0: if info = i, and i is

• <= N: D(i,i) is exactly zero. The factorization has been completed but D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.

• = N+1: D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision.