sysv_rk#

Functions

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

SSYSV_RK 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 bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A = P*U*D*(U**T)*(P**T), if UPLO = ‘U’, or A = P*L*D*(L**T)*(P**T), if UPLO = ‘L’, where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

ssytrf_rk is called to compute the factorization of a real symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine ssytrs_3.

  • Only diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k)=A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

For more info see the description of ssytrf_rk

.

For a 1-by-1 diagonal block

D(k), the element E[k] is set to 0 in both uplo='U' or uplo='L' cases.

For more info see the description of ssytrf_rk

.

For more info see the description of

ssytrf_rk.

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, diagonal of the block diagonal matrix D and factors U or L as computed by ssytrf_rk:

in
lda

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

out
E

Array of dimension n. On exit, contains the output computed by the factorization routine ssytrf_rk, i.e. the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where if uplo='U': E[i]=D(i-1,i), i=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0.

out
ipiv

Array of dimension n. Details of the interchanges and the block structure of D, as determined by ssytrf_rk.

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). Work array used in the factorization stage. On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance of the factorization stage lwork>=n*nb, where nb is the optimal block size for ssytrf_rk. If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array for the factorization stage, returns this value as the first entry of the work array, and no error message related to lwork is issued.

out
info

  • info=0: successful exit

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

  • info>0: if info=k, the matrix A is singular, because column k in the upper (uplo='U') or lower (uplo='L') triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Functions

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

DSYSV_RK 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 bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A = P*U*D*(U**T)*(P**T), if UPLO = ‘U’, or A = P*L*D*(L**T)*(P**T), if UPLO = ‘L’, where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

dsytrf_rk is called to compute the factorization of a real symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine dsytrs_3.

  • Only diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k)=A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

For more info see the description of dsytrf_rk

.

For a 1-by-1 diagonal block

D(k), the element E[k] is set to 0 in both uplo='U' or uplo='L' cases.

For more info see the description of dsytrf_rk

.

For more info see the description of

dsytrf_rk.

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, diagonal of the block diagonal matrix D and factors U or L as computed by dsytrf_rk:

in
lda

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

out
E

Array of dimension n. On exit, contains the output computed by the factorization routine dsytrf_rk, i.e. the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where if uplo='U': E[i]=D(i-1,i), i=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0.

out
ipiv

Array of dimension n. Details of the interchanges and the block structure of D, as determined by dsytrf_rk.

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). Work array used in the factorization stage. On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance of the factorization stage lwork>=n*nb, where nb is the optimal block size for dsytrf_rk. If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array for the factorization stage, returns this value as the first entry of the work array, and no error message related to lwork is issued.

out
info

  • info=0: successful exit

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

  • info>0: if info=k, the matrix A is singular, because column k in the upper (uplo='U') or lower (uplo='L') triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Functions

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

CSYSV_RK 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 bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A = P*U*D*(U**T)*(P**T), if UPLO = ‘U’, or A = P*L*D*(L**T)*(P**T), if UPLO = ‘L’, where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

csytrf_rk is called to compute the factorization of a complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine csytrs_3.

  • Only diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k)=A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

For more info see the description of csytrf_rk

.

For a 1-by-1 diagonal block

D(k), the element E[k] is set to 0 in both uplo='U' or uplo='L' cases.

For more info see the description of csytrf_rk

.

For more info see the description of

csytrf_rk.

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, diagonal of the block diagonal matrix D and factors U or L as computed by csytrf_rk:

in
lda

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

out
E

Complex array of dimension n. On exit, contains the output computed by the factorization routine csytrf_rk, i.e. the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where if uplo='U': E[i]=D(i-1,i), i=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0.

out
ipiv

Array of dimension n. Details of the interchanges and the block structure of D, as determined by csytrf_rk.

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). Work array used in the factorization stage. On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance of the factorization stage lwork>=n*nb, where nb is the optimal block size for csytrf_rk. If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array for the factorization stage, returns this value as the first entry of the work array, and no error message related to lwork is issued.

out
info

  • info=0: successful exit

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

  • info>0: if info=k, the matrix A is singular, because column k in the upper (uplo='U') or lower (uplo='L') triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Functions

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

ZSYSV_RK 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 bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A = P*U*D*(U**T)*(P**T), if UPLO = ‘U’, or A = P*L*D*(L**T)*(P**T), if UPLO = ‘L’, where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

zsytrf_rk is called to compute the factorization of a complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine zsytrs_3.

  • Only diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k)=A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

For more info see the description of zsytrf_rk

.

For a 1-by-1 diagonal block

D(k), the element E[k] is set to 0 in both uplo='U' or uplo='L' cases.

For more info see the description of zsytrf_rk

.

For more info see the description of

zsytrf_rk.

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, diagonal of the block diagonal matrix D and factors U or L as computed by zsytrf_rk:

in
lda

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

out
E

Complex array of dimension n. On exit, contains the output computed by the factorization routine zsytrf_rk, i.e. the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where if uplo='U': E[i]=D(i-1,i), i=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0.

out
ipiv

Array of dimension n. Details of the interchanges and the block structure of D, as determined by zsytrf_rk.

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). Work array used in the factorization stage. On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance of the factorization stage lwork>=n*nb, where nb is the optimal block size for zsytrf_rk. If lwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the work array for the factorization stage, returns this value as the first entry of the work array, and no error message related to lwork is issued.

out
info

  • info=0: successful exit

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

  • info>0: if info=k, the matrix A is singular, because column k in the upper (uplo='U') or lower (uplo='L') triangular part of A contains all zeros. Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.