lasyf_rk

slasyf_rk

Functions

void slasyf_rk(
    const char*          uplo,
    const INT            n,
    const INT            nb,
          INT*           kb,
          f32*  restrict A,
    const INT            lda,
          f32*  restrict E,
          INT*  restrict ipiv,
          f32*  restrict W,
    const INT            ldw,
          INT*           info
);

void slasyf_rk(const char *uplo, const INT n, const INT nb, INT *kb, f32 *restrict A, const INT lda, f32 *restrict E, INT *restrict ipiv, f32 *restrict W, const INT ldw, INT *info)

SLASYF_RK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method.

The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

SLASYF_RK is an auxiliary routine called by SSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = ‘U’) or A22 (if UPLO = ‘L’).

On exit, contains:

• 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 a 1-by-1 diagonal block

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

  cases.

  If

  uplo='U' (in factorization order, k decreases from n-1 to 0):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,n-kb:n-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k-1]<0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the matrix A(0:n-1,n-kb:n-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k-1]-1!=k-1, rows and columns k-1 and -ipiv[k-1]-1 were interchanged in the submatrix A(0:n-1,n-kb:n-1); if -ipiv[k-1]-1=k-1, no interchange occurred.

• In both cases, the recovered partner index is always <=k.

If uplo='L' (in factorization order, k increases from 0 to n-1):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,0:kb-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k+1]<0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k+1]-1!=k+1, rows and columns k+1 and -ipiv[k+1]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k+1]-1=k+1, no interchange occurred.

• In both cases, the recovered partner index is always >=k.

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

The order of the matrix A. n>=0.

in

nb

The maximum number of columns of the matrix A that should be factored. nb should be at least 2 to allow for 2-by-2 pivot blocks.

out

kb

The number of columns of A that were actually factored. kb is either nb-1 or nb, or n if n<=nb.

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.

in

lda

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

out

E

Array of dimension n. On exit, contains 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. ipiv describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of ipiv[k] (recovered as -ipiv[k]-1 when ipiv[k]<0, or ipiv[k] itself when ipiv[k]>=0) represents the index of the row and column that were interchanged with the k-th row and column. The value of uplo describes the order in which the interchanges were applied. Also, the sign of ipiv[k] represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step.

out

W

Array of dimension (ldw,nb).

in

ldw

The leading dimension of the array W. ldw>=max(1,n).

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.

dlasyf_rk

Functions

void dlasyf_rk(
    const char*          uplo,
    const INT            n,
    const INT            nb,
          INT*           kb,
          f64*  restrict A,
    const INT            lda,
          f64*  restrict E,
          INT*  restrict ipiv,
          f64*  restrict W,
    const INT            ldw,
          INT*           info
);

void dlasyf_rk(const char *uplo, const INT n, const INT nb, INT *kb, f64 *restrict A, const INT lda, f64 *restrict E, INT *restrict ipiv, f64 *restrict W, const INT ldw, INT *info)

DLASYF_RK computes a partial factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method.

The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

DLASYF_RK is an auxiliary routine called by DSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = ‘U’) or A22 (if UPLO = ‘L’).

On exit, contains:

• 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 a 1-by-1 diagonal block

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

  cases.

  If

  uplo='U' (in factorization order, k decreases from n-1 to 0):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,n-kb:n-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k-1]<0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the matrix A(0:n-1,n-kb:n-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k-1]-1!=k-1, rows and columns k-1 and -ipiv[k-1]-1 were interchanged in the submatrix A(0:n-1,n-kb:n-1); if -ipiv[k-1]-1=k-1, no interchange occurred.

• In both cases, the recovered partner index is always <=k.

If uplo='L' (in factorization order, k increases from 0 to n-1):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,0:kb-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k+1]<0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k+1]-1!=k+1, rows and columns k+1 and -ipiv[k+1]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k+1]-1=k+1, no interchange occurred.

• In both cases, the recovered partner index is always >=k.

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

The order of the matrix A. n>=0.

in

nb

The maximum number of columns of the matrix A that should be factored. nb should be at least 2 to allow for 2-by-2 pivot blocks.

out

kb

The number of columns of A that were actually factored. kb is either nb-1 or nb, or n if n<=nb.

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.

in

lda

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

out

E

Array of dimension n. On exit, contains 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. ipiv describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of ipiv[k] (recovered as -ipiv[k]-1 when ipiv[k]<0, or ipiv[k] itself when ipiv[k]>=0) represents the index of the row and column that were interchanged with the k-th row and column. The value of uplo describes the order in which the interchanges were applied. Also, the sign of ipiv[k] represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step.

out

W

Array of dimension (ldw,nb).

in

ldw

The leading dimension of the array W. ldw>=max(1,n).

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.

clasyf_rk

Functions

void clasyf_rk(
    const char*          uplo,
    const INT            n,
    const INT            nb,
          INT*           kb,
          c64*  restrict A,
    const INT            lda,
          c64*  restrict E,
          INT*  restrict ipiv,
          c64*  restrict W,
    const INT            ldw,
          INT*           info
);

void clasyf_rk(const char *uplo, const INT n, const INT nb, INT *kb, c64 *restrict A, const INT lda, c64 *restrict E, INT *restrict ipiv, c64 *restrict W, const INT ldw, INT *info)

CLASYF_RK computes a partial factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method.

The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

CLASYF_RK is an auxiliary routine called by CSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = ‘U’) or A22 (if UPLO = ‘L’).

On exit, contains:

• 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 a 1-by-1 diagonal block

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

  cases.

  If

  uplo='U' (in factorization order, k decreases from n-1 to 0):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,n-kb:n-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k-1]<0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the matrix A(0:n-1,n-kb:n-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k-1]-1!=k-1, rows and columns k-1 and -ipiv[k-1]-1 were interchanged in the submatrix A(0:n-1,n-kb:n-1); if -ipiv[k-1]-1=k-1, no interchange occurred.

• In both cases, the recovered partner index is always <=k.

If uplo='L' (in factorization order, k increases from 0 to n-1):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,0:kb-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k+1]<0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k+1]-1!=k+1, rows and columns k+1 and -ipiv[k+1]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k+1]-1=k+1, no interchange occurred.

• In both cases, the recovered partner index is always >=k.

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

The order of the matrix A. n>=0.

in

nb

The maximum number of columns of the matrix A that should be factored. nb should be at least 2 to allow for 2-by-2 pivot blocks.

out

kb

The number of columns of A that were actually factored. kb is either nb-1 or nb, or n if n<=nb.

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.

in

lda

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

out

E

Complex array of dimension n. On exit, contains 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. ipiv describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of ipiv[k] (recovered as -ipiv[k]-1 when ipiv[k]<0, or ipiv[k] itself when ipiv[k]>=0) represents the index of the row and column that were interchanged with the k-th row and column. The value of uplo describes the order in which the interchanges were applied. Also, the sign of ipiv[k] represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step.

out

W

Complex array of dimension (ldw,nb).

in

ldw

The leading dimension of the array W. ldw>=max(1,n).

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.

zlasyf_rk

Functions

void zlasyf_rk(
    const char*          uplo,
    const INT            n,
    const INT            nb,
          INT*           kb,
          c128* restrict A,
    const INT            lda,
          c128* restrict E,
          INT*  restrict ipiv,
          c128* restrict W,
    const INT            ldw,
          INT*           info
);

void zlasyf_rk(const char *uplo, const INT n, const INT nb, INT *kb, c128 *restrict A, const INT lda, c128 *restrict E, INT *restrict ipiv, c128 *restrict W, const INT ldw, INT *info)

ZLASYF_RK computes a partial factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method.

The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

ZLASYF_RK is an auxiliary routine called by ZSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = ‘U’) or A22 (if UPLO = ‘L’).

On exit, contains:

• 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 a 1-by-1 diagonal block

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

  cases.

  If

  uplo='U' (in factorization order, k decreases from n-1 to 0):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,n-kb:n-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k-1]<0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the matrix A(0:n-1,n-kb:n-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k-1]-1!=k-1, rows and columns k-1 and -ipiv[k-1]-1 were interchanged in the submatrix A(0:n-1,n-kb:n-1); if -ipiv[k-1]-1=k-1, no interchange occurred.

• In both cases, the recovered partner index is always <=k.

If uplo='L' (in factorization order, k increases from 0 to n-1):

• A single non-negative entry ipiv[k]>=0 means: D(k,k) is a 1-by-1 diagonal block. If ipiv[k]!=k, rows and columns k and ipiv[k] were interchanged in the submatrix A(0:n-1,0:kb-1); if ipiv[k]=k, no interchange occurred.

• A pair of consecutive negative entries ipiv[k]<0 and ipiv[k+1]<0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Negative entries in ipiv appear only in pairs.) 1) If -ipiv[k]-1!=k, rows and columns k and -ipiv[k]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k]-1=k, no interchange occurred. 2) If -ipiv[k+1]-1!=k+1, rows and columns k+1 and -ipiv[k+1]-1 were interchanged in the submatrix A(0:n-1,0:kb-1); if -ipiv[k+1]-1=k+1, no interchange occurred.

• In both cases, the recovered partner index is always >=k.

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

The order of the matrix A. n>=0.

in

nb

The maximum number of columns of the matrix A that should be factored. nb should be at least 2 to allow for 2-by-2 pivot blocks.

out

kb

The number of columns of A that were actually factored. kb is either nb-1 or nb, or n if n<=nb.

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.

in

lda

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

out

E

Complex array of dimension n. On exit, contains 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. ipiv describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of ipiv[k] (recovered as -ipiv[k]-1 when ipiv[k]<0, or ipiv[k] itself when ipiv[k]>=0) represents the index of the row and column that were interchanged with the k-th row and column. The value of uplo describes the order in which the interchanges were applied. Also, the sign of ipiv[k] represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step.

out

W

Complex array of dimension (ldw,nb).

in

ldw

The leading dimension of the array W. ldw>=max(1,n).

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.