sytf2_rook

ssytf2_rook

Functions

void ssytf2_rook(
    const char*          uplo,
    const INT            n,
          f32*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          INT*           info
);

void ssytf2_rook(const char *uplo, const INT n, f32 *restrict A, const INT lda, INT *restrict ipiv, INT *info)

SSYTF2_ROOK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method:

A = U*D*U**T  or  A = L*D*L**T

This is the unblocked version of the algorithm, calling Level 2 BLAS.

• 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.

  Further Details:

  If uplo='U', then A = U*D*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    v    0   )   k-s+1
  U(k) =  (   0    I    0   )   s
          (   0    0    I   )   n-1-k
             k-s+1  s   n-1-k
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(0:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(0:k-2,k-1:k).

  If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    0     0   )  k
  L(k) =  (   0    I     0   )  s
          (   0    v     I   )  n-k-s
             k     s   n-k-s
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n-1,k:k+1).

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

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

inout

A

Array of dimension (lda,n). On entry, the symmetric matrix A. On exit, the block diagonal matrix D and the multipliers (see below for further details).

in

lda

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

out

ipiv

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

out

info

• info=0: successful exit

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

• info>0: if info=k, D(k,k) is exactly zero. The factorization has been completed, but D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

dsytf2_rook

Functions

void dsytf2_rook(
    const char*          uplo,
    const INT            n,
          f64*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          INT*           info
);

void dsytf2_rook(const char *uplo, const INT n, f64 *restrict A, const INT lda, INT *restrict ipiv, INT *info)

DSYTF2_ROOK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method:

A = U*D*U**T  or  A = L*D*L**T

This is the unblocked version of the algorithm, calling Level 2 BLAS.

• 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.

  Further Details:

  If uplo='U', then A = U*D*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    v    0   )   k-s+1
  U(k) =  (   0    I    0   )   s
          (   0    0    I   )   n-1-k
             k-s+1  s   n-1-k
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(0:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(0:k-2,k-1:k).

  If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    0     0   )  k
  L(k) =  (   0    I     0   )  s
          (   0    v     I   )  n-k-s
             k     s   n-k-s
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n-1,k:k+1).

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

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

inout

A

Array of dimension (lda,n). On entry, the symmetric matrix A. On exit, the block diagonal matrix D and the multipliers (see below for further details).

in

lda

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

out

ipiv

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

out

info

• info=0: successful exit

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

• info>0: if info=k, D(k,k) is exactly zero. The factorization has been completed, but D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

csytf2_rook

Functions

void csytf2_rook(
    const char*          uplo,
    const INT            n,
          c64*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          INT*           info
);

void csytf2_rook(const char *uplo, const INT n, c64 *restrict A, const INT lda, INT *restrict ipiv, INT *info)

CSYTF2_ROOK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method:

A = U*D*U**T  or  A = L*D*L**T

This is the unblocked version of the algorithm, calling Level 2 BLAS.

• 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.

  Further Details:

  If uplo='U', then A = U*D*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    v    0   )   k-s+1
  U(k) =  (   0    I    0   )   s
          (   0    0    I   )   n-1-k
             k-s+1  s   n-1-k
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(0:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(0:k-2,k-1:k).

  If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    0     0   )  k
  L(k) =  (   0    I     0   )  s
          (   0    v     I   )  n-k-s
             k     s   n-k-s
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n-1,k:k+1).

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

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

inout

A

Complex array of dimension (lda,n). On entry, the symmetric matrix A. On exit, the block diagonal matrix D and the multipliers (see below for further details).

in

lda

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

out

ipiv

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

out

info

• info=0: successful exit

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

• info>0: if info=k, D(k,k) is exactly zero. The factorization has been completed, but D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

zsytf2_rook

Functions

void zsytf2_rook(
    const char*          uplo,
    const INT            n,
          c128* restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          INT*           info
);

void zsytf2_rook(const char *uplo, const INT n, c128 *restrict A, const INT lda, INT *restrict ipiv, INT *info)

ZSYTF2_ROOK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (“rook”) diagonal pivoting method:

A = U*D*U**T  or  A = L*D*L**T

This is the unblocked version of the algorithm, calling Level 2 BLAS.

• 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.

  Further Details:

  If uplo='U', then A = U*D*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    v    0   )   k-s+1
  U(k) =  (   0    I    0   )   s
          (   0    0    I   )   n-1-k
             k-s+1  s   n-1-k
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(0:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(0:k-2,k-1:k).

  If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by ipiv[k], and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

          (   I    0     0   )  k
  L(k) =  (   0    I     0   )  s
          (   0    v     I   )  n-k-s
             k     s   n-k-s
  

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n-1,k:k+1).

Parameters

in

uplo

• 'U': Upper triangular

• 'L': Lower triangular

in

n

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

inout

A

Complex array of dimension (lda,n). On entry, the symmetric matrix A. On exit, the block diagonal matrix D and the multipliers (see below for further details).

in

lda

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

out

ipiv

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

out

info

• info=0: successful exit

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

• info>0: if info=k, D(k,k) is exactly zero. The factorization has been completed, but D is exactly singular, and division by zero will occur if it is used to solve a system of equations.