--- rpl/lapack/lapack/zhptrf.f 2010/08/07 13:22:35 1.5 +++ rpl/lapack/lapack/zhptrf.f 2018/05/29 07:18:22 1.17 @@ -1,9 +1,168 @@ +*> \brief \b ZHPTRF +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZHPTRF + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZHPTRF( UPLO, N, AP, IPIV, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER UPLO +* INTEGER INFO, N +* .. +* .. Array Arguments .. +* INTEGER IPIV( * ) +* COMPLEX*16 AP( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> ZHPTRF computes the factorization of a complex Hermitian packed +*> matrix A using the Bunch-Kaufman diagonal pivoting method: +*> +*> A = U*D*U**H or A = L*D*L**H +*> +*> where U (or L) is a product of permutation and unit upper (lower) +*> triangular matrices, and D is Hermitian and block diagonal with +*> 1-by-1 and 2-by-2 diagonal blocks. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] UPLO +*> \verbatim +*> UPLO is CHARACTER*1 +*> = 'U': Upper triangle of A is stored; +*> = 'L': Lower triangle of A is stored. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in,out] AP +*> \verbatim +*> AP is COMPLEX*16 array, dimension (N*(N+1)/2) +*> On entry, the upper or lower triangle of the Hermitian matrix +*> A, packed columnwise in a linear array. The j-th column of A +*> is stored in the array AP as follows: +*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; +*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. +*> +*> On exit, the block diagonal matrix D and the multipliers used +*> to obtain the factor U or L, stored as a packed triangular +*> matrix overwriting A (see below for further details). +*> \endverbatim +*> +*> \param[out] IPIV +*> \verbatim +*> IPIV is INTEGER array, dimension (N) +*> Details of the interchanges and the block structure of D. +*> 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 UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and +*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) +*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = +*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were +*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -i, the i-th argument had an illegal value +*> > 0: if INFO = i, D(i,i) is exactly zero. 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. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date December 2016 +* +*> \ingroup complex16OTHERcomputational +* +*> \par Further Details: +* ===================== +*> +*> \verbatim +*> +*> If UPLO = 'U', then A = U*D*U**H, where +*> U = P(n)*U(n)* ... *P(k)U(k)* ..., +*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to +*> 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 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 +*> U(k) = ( 0 I 0 ) s +*> ( 0 0 I ) n-k +*> k-s s n-k +*> +*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1: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(1:k-2,k-1:k). +*> +*> If UPLO = 'L', then A = L*D*L**H, where +*> L = P(1)*L(1)* ... *P(k)*L(k)* ..., +*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to +*> n 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-1 +*> L(k) = ( 0 I 0 ) s +*> ( 0 v I ) n-k-s+1 +*> k-1 s n-k-s+1 +*> +*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,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,k:k+1). +*> \endverbatim +* +*> \par Contributors: +* ================== +*> +*> J. Lewis, Boeing Computer Services Company +* +* ===================================================================== SUBROUTINE ZHPTRF( UPLO, N, AP, IPIV, INFO ) * -* -- LAPACK routine (version 3.2) -- +* -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 +* December 2016 * * .. Scalar Arguments .. CHARACTER UPLO @@ -14,97 +173,6 @@ COMPLEX*16 AP( * ) * .. * -* Purpose -* ======= -* -* ZHPTRF computes the factorization of a complex Hermitian packed -* matrix A using the Bunch-Kaufman diagonal pivoting method: -* -* A = U*D*U**H or A = L*D*L**H -* -* where U (or L) is a product of permutation and unit upper (lower) -* triangular matrices, and D is Hermitian and block diagonal with -* 1-by-1 and 2-by-2 diagonal blocks. -* -* Arguments -* ========= -* -* UPLO (input) CHARACTER*1 -* = 'U': Upper triangle of A is stored; -* = 'L': Lower triangle of A is stored. -* -* N (input) INTEGER -* The order of the matrix A. N >= 0. -* -* AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2) -* On entry, the upper or lower triangle of the Hermitian matrix -* A, packed columnwise in a linear array. The j-th column of A -* is stored in the array AP as follows: -* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; -* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. -* -* On exit, the block diagonal matrix D and the multipliers used -* to obtain the factor U or L, stored as a packed triangular -* matrix overwriting A (see below for further details). -* -* IPIV (output) INTEGER array, dimension (N) -* Details of the interchanges and the block structure of D. -* 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 UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and -* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) -* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = -* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were -* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. -* -* INFO (output) INTEGER -* = 0: successful exit -* < 0: if INFO = -i, the i-th argument had an illegal value -* > 0: if INFO = i, D(i,i) is exactly zero. 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. -* -* Further Details -* =============== -* -* 5-96 - Based on modifications by J. Lewis, Boeing Computer Services -* Company -* -* If UPLO = 'U', then A = U*D*U', where -* U = P(n)*U(n)* ... *P(k)U(k)* ..., -* i.e., U is a product of terms P(k)*U(k), where k decreases from n to -* 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 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 -* U(k) = ( 0 I 0 ) s -* ( 0 0 I ) n-k -* k-s s n-k -* -* If s = 1, D(k) overwrites A(k,k), and v overwrites A(1: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(1:k-2,k-1:k). -* -* If UPLO = 'L', then A = L*D*L', where -* L = P(1)*L(1)* ... *P(k)*L(k)* ..., -* i.e., L is a product of terms P(k)*L(k), where k increases from 1 to -* n 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-1 -* L(k) = ( 0 I 0 ) s -* ( 0 v I ) n-k-s+1 -* k-1 s n-k-s+1 -* -* If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,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,k:k+1). -* * ===================================================================== * * .. Parameters .. @@ -161,7 +229,7 @@ * IF( UPPER ) THEN * -* Factorize A as U*D*U' using the upper triangle of A +* Factorize A as U*D*U**H using the upper triangle of A * * K is the main loop index, decreasing from N to 1 in steps of * 1 or 2 @@ -293,7 +361,7 @@ * * Perform a rank-1 update of A(1:k-1,1:k-1) as * -* A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' +* A := A - U(k)*D(k)*U(k)**H = A - W(k)*1/D(k)*W(k)**H * R1 = ONE / DBLE( AP( KC+K-1 ) ) CALL ZHPR( UPLO, K-1, -R1, AP( KC ), 1, AP ) @@ -312,8 +380,8 @@ * * Perform a rank-2 update of A(1:k-2,1:k-2) as * -* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )' -* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' +* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**H +* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**H * IF( K.GT.2 ) THEN * @@ -363,7 +431,7 @@ * ELSE * -* Factorize A as L*D*L' using the lower triangle of A +* Factorize A as L*D*L**H using the lower triangle of A * * K is the main loop index, increasing from 1 to N in steps of * 1 or 2 @@ -498,7 +566,7 @@ * * Perform a rank-1 update of A(k+1:n,k+1:n) as * -* A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' +* A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H * R1 = ONE / DBLE( AP( KC ) ) CALL ZHPR( UPLO, N-K, -R1, AP( KC+1 ), 1, @@ -521,8 +589,8 @@ * * Perform a rank-2 update of A(k+2:n,k+2:n) as * -* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' -* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )' +* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H +* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H * * where L(k) and L(k+1) are the k-th and (k+1)-th * columns of L