.TH "lahef_rk" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME lahef_rk \- la{he,sy}f_rk: triangular factor step .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBclahef_rk\fP (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)" .br .RI "\fBCLAHEF_RK\fP computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. " .ti -1c .RI "subroutine \fBclasyf_rk\fP (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)" .br .RI "\fBCLASYF_RK\fP computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. " .ti -1c .RI "subroutine \fBdlasyf_rk\fP (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)" .br .RI "\fBDLASYF_RK\fP computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. " .ti -1c .RI "subroutine \fBslasyf_rk\fP (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)" .br .RI "\fBSLASYF_RK\fP computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. " .ti -1c .RI "subroutine \fBzlahef_rk\fP (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)" .br .RI "\fBZLAHEF_RK\fP computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. " .ti -1c .RI "subroutine \fBzlasyf_rk\fP (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)" .br .RI "\fBZLASYF_RK\fP computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine clahef_rk (character uplo, integer n, integer nb, integer kb, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( ldw, * ) w, integer ldw, integer info)" .PP \fBCLAHEF_RK\fP computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CLAHEF_RK computes a partial factorization of a complex Hermitian 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**H U22**H ) A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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\&. CLAHEF_RK is an auxiliary routine called by CHETRF_RK\&. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L')\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER 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\&. .fi .PP .br \fIKB\fP .PP .nf KB is INTEGER The number of columns of A that were actually factored\&. KB is either NB-1 or NB, or N if N <= NB\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX array, dimension (LDA,N) On entry, the Hermitian 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, contains: a) ONLY diagonal elements of the Hermitian 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 b) If UPLO = 'U': factor U in the superdiagonal part of A\&. If UPLO = 'L': factor L in the subdiagonal part of A\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is COMPLEX array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0\&. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases\&. .fi .PP .br \fIIPIV\fP .PP .nf IPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows\&. The absolute value of IPIV(k) represents the index of 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 represents the block structure of the Hermitian 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\&. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive 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(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N)\&. If -IPIV(k-1) = k-1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive 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(1:N,1:KB)\&. If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k+1) = k+1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. .fi .PP .br \fIW\fP .PP .nf W is COMPLEX array, dimension (LDW,NB) .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,N)\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros\&. If UPLO = 'L': column k in the lower 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\&. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J\&. Higham, Craig Lucas, School of Mathematics, University of Manchester .fi .PP .RE .PP .SS "subroutine clasyf_rk (character uplo, integer n, integer nb, integer kb, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( ldw, * ) w, integer ldw, integer info)" .PP \fBCLASYF_RK\fP computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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')\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER 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\&. .fi .PP .br \fIKB\fP .PP .nf KB is INTEGER The number of columns of A that were actually factored\&. KB is either NB-1 or NB, or N if N <= NB\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX array, 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, contains: a) 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 b) If UPLO = 'U': factor U in the superdiagonal part of A\&. If UPLO = 'L': factor L in the subdiagonal part of A\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is COMPLEX array, 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=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0\&. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases\&. .fi .PP .br \fIIPIV\fP .PP .nf IPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows\&. The absolute value of IPIV(k) represents the index of 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 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\&. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive 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(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N)\&. If -IPIV(k-1) = k-1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive 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(1:N,1:KB)\&. If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k+1) = k+1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. .fi .PP .br \fIW\fP .PP .nf W is COMPLEX array, dimension (LDW,NB) .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,N)\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros\&. If UPLO = 'L': column k in the lower 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\&. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J\&. Higham, Craig Lucas, School of Mathematics, University of Manchester .fi .PP .RE .PP .SS "subroutine dlasyf_rk (character uplo, integer n, integer nb, integer kb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, integer, dimension( * ) ipiv, double precision, dimension( ldw, * ) w, integer ldw, integer info)" .PP \fBDLASYF_RK\fP computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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')\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER 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\&. .fi .PP .br \fIKB\fP .PP .nf KB is INTEGER The number of columns of A that were actually factored\&. KB is either NB-1 or NB, or N if N <= NB\&. .fi .PP .br \fIA\fP .PP .nf A is DOUBLE PRECISION array, 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, contains: a) 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 b) If UPLO = 'U': factor U in the superdiagonal part of A\&. If UPLO = 'L': factor L in the subdiagonal part of A\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is DOUBLE PRECISION array, 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=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0\&. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases\&. .fi .PP .br \fIIPIV\fP .PP .nf IPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows\&. The absolute value of IPIV(k) represents the index of 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 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\&. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive 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(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N)\&. If -IPIV(k-1) = k-1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive 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(1:N,1:KB)\&. If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k+1) = k+1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. .fi .PP .br \fIW\fP .PP .nf W is DOUBLE PRECISION array, dimension (LDW,NB) .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,N)\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros\&. If UPLO = 'L': column k in the lower 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\&. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J\&. Higham, Craig Lucas, School of Mathematics, University of Manchester .fi .PP .RE .PP .SS "subroutine slasyf_rk (character uplo, integer n, integer nb, integer kb, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, real, dimension( ldw, * ) w, integer ldw, integer info)" .PP \fBSLASYF_RK\fP computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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')\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER 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\&. .fi .PP .br \fIKB\fP .PP .nf KB is INTEGER The number of columns of A that were actually factored\&. KB is either NB-1 or NB, or N if N <= NB\&. .fi .PP .br \fIA\fP .PP .nf A is REAL array, 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, contains: a) 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 b) If UPLO = 'U': factor U in the superdiagonal part of A\&. If UPLO = 'L': factor L in the subdiagonal part of A\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is REAL array, 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=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0\&. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases\&. .fi .PP .br \fIIPIV\fP .PP .nf IPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows\&. The absolute value of IPIV(k) represents the index of 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 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\&. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive 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(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N)\&. If -IPIV(k-1) = k-1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive 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(1:N,1:KB)\&. If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k+1) = k+1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. .fi .PP .br \fIW\fP .PP .nf W is REAL array, dimension (LDW,NB) .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,N)\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros\&. If UPLO = 'L': column k in the lower 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\&. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J\&. Higham, Craig Lucas, School of Mathematics, University of Manchester .fi .PP .RE .PP .SS "subroutine zlahef_rk (character uplo, integer n, integer nb, integer kb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( ldw, * ) w, integer ldw, integer info)" .PP \fBZLAHEF_RK\fP computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. .PP \fBPurpose:\fP .RS 4 .PP .nf ZLAHEF_RK computes a partial factorization of a complex Hermitian 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**H U22**H ) A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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\&. ZLAHEF_RK is an auxiliary routine called by ZHETRF_RK\&. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L')\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER 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\&. .fi .PP .br \fIKB\fP .PP .nf KB is INTEGER The number of columns of A that were actually factored\&. KB is either NB-1 or NB, or N if N <= NB\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, dimension (LDA,N) On entry, the Hermitian 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, contains: a) ONLY diagonal elements of the Hermitian 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 b) If UPLO = 'U': factor U in the superdiagonal part of A\&. If UPLO = 'L': factor L in the subdiagonal part of A\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is COMPLEX*16 array, dimension (N) On exit, contains the superdiagonal (or subdiagonal) elements of the Hermitian 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=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0\&. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases\&. .fi .PP .br \fIIPIV\fP .PP .nf IPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows\&. The absolute value of IPIV(k) represents the index of 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 represents the block structure of the Hermitian 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\&. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive 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(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N)\&. If -IPIV(k-1) = k-1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive 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(1:N,1:KB)\&. If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k+1) = k+1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. .fi .PP .br \fIW\fP .PP .nf W is COMPLEX*16 array, dimension (LDW,NB) .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,N)\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros\&. If UPLO = 'L': column k in the lower 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\&. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J\&. Higham, Craig Lucas, School of Mathematics, University of Manchester .fi .PP .RE .PP .SS "subroutine zlasyf_rk (character uplo, integer n, integer nb, integer kb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( ldw, * ) w, integer ldw, integer info)" .PP \fBZLASYF_RK\fP computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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')\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER 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\&. .fi .PP .br \fIKB\fP .PP .nf KB is INTEGER The number of columns of A that were actually factored\&. KB is either NB-1 or NB, or N if N <= NB\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, 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, contains: a) 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 b) If UPLO = 'U': factor U in the superdiagonal part of A\&. If UPLO = 'L': factor L in the subdiagonal part of A\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is COMPLEX*16 array, 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=2:N, E(1) is set to 0; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0\&. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases\&. .fi .PP .br \fIIPIV\fP .PP .nf IPIV is INTEGER array, dimension (N) IPIV describes the permutation matrix P in the factorization of matrix A as follows\&. The absolute value of IPIV(k) represents the index of 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 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\&. If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive 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(1:N,N-KB+1:N); If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N)\&. If -IPIV(k-1) = k-1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) <= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. If UPLO = 'L', ( in factorization order, k increases from 1 to N ): a) A single positive 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(1:N,1:KB)\&. If IPIV(k) = k, no interchange occurred\&. b) 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\&. (NOTE: negative entries in IPIV appear ONLY in pairs)\&. 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k) = k, no interchange occurred\&. 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB)\&. If -IPIV(k+1) = k+1, no interchange occurred\&. c) In both cases a) and b) is always ABS( IPIV(k) ) >= k\&. d) NOTE: Any entry IPIV(k) is always NONZERO on output\&. .fi .PP .br \fIW\fP .PP .nf W is COMPLEX*16 array, dimension (LDW,NB) .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,N)\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: If INFO = -k, the k-th argument had an illegal value > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros\&. If UPLO = 'L': column k in the lower 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\&. NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley September 2007, Sven Hammarling, Nicholas J\&. Higham, Craig Lucas, School of Mathematics, University of Manchester .fi .PP .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.