sytrf_rk#

Functions

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

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

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

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.

This is the blocked version of the algorithm, calling Level 3 BLAS.

  • 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.

  • 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.

Parameters

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. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

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

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, contains:

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. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

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 of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or 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. info only stores the first occurrence of a singularity; the factorization always completes.

Functions

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

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

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

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.

This is the blocked version of the algorithm, calling Level 3 BLAS.

  • 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.

  • 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.

Parameters

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. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

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

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, contains:

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. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

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 of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or 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. info only stores the first occurrence of a singularity; the factorization always completes.

Functions

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

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

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

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.

This is the blocked version of the algorithm, calling Level 3 BLAS.

  • 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.

  • 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.

Parameters

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. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

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

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, contains:

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. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

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 of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or 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. info only stores the first occurrence of a singularity; the factorization always completes.

Functions

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

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

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

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.

This is the blocked version of the algorithm, calling Level 3 BLAS.

  • 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.

  • 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.

Parameters

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. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

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

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, contains:

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. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

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 of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or 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. info only stores the first occurrence of a singularity; the factorization always completes.