Annotation of rpl/lapack/lapack/zhecon_3.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b ZHECON_3
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZHECON_3 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhecon_3.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhecon_3.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhecon_3.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZHECON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
! 22: * WORK, IWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER UPLO
! 26: * INTEGER INFO, LDA, N
! 27: * DOUBLE PRECISION ANORM, RCOND
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * INTEGER IPIV( * ), IWORK( * )
! 31: * COMPLEX*16 A( LDA, * ), E ( * ), WORK( * )
! 32: * ..
! 33: *
! 34: *
! 35: *> \par Purpose:
! 36: * =============
! 37: *>
! 38: *> \verbatim
! 39: *> ZHECON_3 estimates the reciprocal of the condition number (in the
! 40: *> 1-norm) of a complex Hermitian matrix A using the factorization
! 41: *> computed by ZHETRF_RK or ZHETRF_BK:
! 42: *>
! 43: *> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
! 44: *>
! 45: *> where U (or L) is unit upper (or lower) triangular matrix,
! 46: *> U**H (or L**H) is the conjugate of U (or L), P is a permutation
! 47: *> matrix, P**T is the transpose of P, and D is Hermitian and block
! 48: *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
! 49: *>
! 50: *> An estimate is obtained for norm(inv(A)), and the reciprocal of the
! 51: *> condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
! 52: *> This routine uses BLAS3 solver ZHETRS_3.
! 53: *> \endverbatim
! 54: *
! 55: * Arguments:
! 56: * ==========
! 57: *
! 58: *> \param[in] UPLO
! 59: *> \verbatim
! 60: *> UPLO is CHARACTER*1
! 61: *> Specifies whether the details of the factorization are
! 62: *> stored as an upper or lower triangular matrix:
! 63: *> = 'U': Upper triangular, form is A = P*U*D*(U**H)*(P**T);
! 64: *> = 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).
! 65: *> \endverbatim
! 66: *>
! 67: *> \param[in] N
! 68: *> \verbatim
! 69: *> N is INTEGER
! 70: *> The order of the matrix A. N >= 0.
! 71: *> \endverbatim
! 72: *>
! 73: *> \param[in] A
! 74: *> \verbatim
! 75: *> A is COMPLEX*16 array, dimension (LDA,N)
! 76: *> Diagonal of the block diagonal matrix D and factors U or L
! 77: *> as computed by ZHETRF_RK and ZHETRF_BK:
! 78: *> a) ONLY diagonal elements of the Hermitian block diagonal
! 79: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
! 80: *> (superdiagonal (or subdiagonal) elements of D
! 81: *> should be provided on entry in array E), and
! 82: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
! 83: *> If UPLO = 'L': factor L in the subdiagonal part of A.
! 84: *> \endverbatim
! 85: *>
! 86: *> \param[in] LDA
! 87: *> \verbatim
! 88: *> LDA is INTEGER
! 89: *> The leading dimension of the array A. LDA >= max(1,N).
! 90: *> \endverbatim
! 91: *>
! 92: *> \param[in] E
! 93: *> \verbatim
! 94: *> E is COMPLEX*16 array, dimension (N)
! 95: *> On entry, contains the superdiagonal (or subdiagonal)
! 96: *> elements of the Hermitian block diagonal matrix D
! 97: *> with 1-by-1 or 2-by-2 diagonal blocks, where
! 98: *> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not refernced;
! 99: *> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
! 100: *>
! 101: *> NOTE: For 1-by-1 diagonal block D(k), where
! 102: *> 1 <= k <= N, the element E(k) is not referenced in both
! 103: *> UPLO = 'U' or UPLO = 'L' cases.
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[in] IPIV
! 107: *> \verbatim
! 108: *> IPIV is INTEGER array, dimension (N)
! 109: *> Details of the interchanges and the block structure of D
! 110: *> as determined by ZHETRF_RK or ZHETRF_BK.
! 111: *> \endverbatim
! 112: *>
! 113: *> \param[in] ANORM
! 114: *> \verbatim
! 115: *> ANORM is DOUBLE PRECISION
! 116: *> The 1-norm of the original matrix A.
! 117: *> \endverbatim
! 118: *>
! 119: *> \param[out] RCOND
! 120: *> \verbatim
! 121: *> RCOND is DOUBLE PRECISION
! 122: *> The reciprocal of the condition number of the matrix A,
! 123: *> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
! 124: *> estimate of the 1-norm of inv(A) computed in this routine.
! 125: *> \endverbatim
! 126: *>
! 127: *> \param[out] WORK
! 128: *> \verbatim
! 129: *> WORK is COMPLEX*16 array, dimension (2*N)
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[out] IWORK
! 133: *> \verbatim
! 134: *> IWORK is INTEGER array, dimension (N)
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[out] INFO
! 138: *> \verbatim
! 139: *> INFO is INTEGER
! 140: *> = 0: successful exit
! 141: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 142: *> \endverbatim
! 143: *
! 144: * Authors:
! 145: * ========
! 146: *
! 147: *> \author Univ. of Tennessee
! 148: *> \author Univ. of California Berkeley
! 149: *> \author Univ. of Colorado Denver
! 150: *> \author NAG Ltd.
! 151: *
! 152: *> \date December 2016
! 153: *
! 154: *> \ingroup complex16HEcomputational
! 155: *
! 156: *> \par Contributors:
! 157: * ==================
! 158: *> \verbatim
! 159: *>
! 160: *> December 2016, Igor Kozachenko,
! 161: *> Computer Science Division,
! 162: *> University of California, Berkeley
! 163: *>
! 164: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
! 165: *> School of Mathematics,
! 166: *> University of Manchester
! 167: *>
! 168: *> \endverbatim
! 169: *
! 170: * =====================================================================
! 171: SUBROUTINE ZHECON_3( UPLO, N, A, LDA, E, IPIV, ANORM, RCOND,
! 172: $ WORK, INFO )
! 173: *
! 174: * -- LAPACK computational routine (version 3.7.0) --
! 175: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 176: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 177: * December 2016
! 178: *
! 179: * .. Scalar Arguments ..
! 180: CHARACTER UPLO
! 181: INTEGER INFO, LDA, N
! 182: DOUBLE PRECISION ANORM, RCOND
! 183: * ..
! 184: * .. Array Arguments ..
! 185: INTEGER IPIV( * )
! 186: COMPLEX*16 A( LDA, * ), E( * ), WORK( * )
! 187: * ..
! 188: *
! 189: * =====================================================================
! 190: *
! 191: * .. Parameters ..
! 192: REAL ONE, ZERO
! 193: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 194: * ..
! 195: * .. Local Scalars ..
! 196: LOGICAL UPPER
! 197: INTEGER I, KASE
! 198: DOUBLE PRECISION AINVNM
! 199: * ..
! 200: * .. Local Arrays ..
! 201: INTEGER ISAVE( 3 )
! 202: * ..
! 203: * .. External Functions ..
! 204: LOGICAL LSAME
! 205: EXTERNAL LSAME
! 206: * ..
! 207: * .. External Subroutines ..
! 208: EXTERNAL ZHETRS_3, ZLACN2, XERBLA
! 209: * ..
! 210: * .. Intrinsic Functions ..
! 211: INTRINSIC MAX
! 212: * ..
! 213: * .. Executable Statements ..
! 214: *
! 215: * Test the input parameters.
! 216: *
! 217: INFO = 0
! 218: UPPER = LSAME( UPLO, 'U' )
! 219: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 220: INFO = -1
! 221: ELSE IF( N.LT.0 ) THEN
! 222: INFO = -2
! 223: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 224: INFO = -4
! 225: ELSE IF( ANORM.LT.ZERO ) THEN
! 226: INFO = -7
! 227: END IF
! 228: IF( INFO.NE.0 ) THEN
! 229: CALL XERBLA( 'ZHECON_3', -INFO )
! 230: RETURN
! 231: END IF
! 232: *
! 233: * Quick return if possible
! 234: *
! 235: RCOND = ZERO
! 236: IF( N.EQ.0 ) THEN
! 237: RCOND = ONE
! 238: RETURN
! 239: ELSE IF( ANORM.LE.ZERO ) THEN
! 240: RETURN
! 241: END IF
! 242: *
! 243: * Check that the diagonal matrix D is nonsingular.
! 244: *
! 245: IF( UPPER ) THEN
! 246: *
! 247: * Upper triangular storage: examine D from bottom to top
! 248: *
! 249: DO I = N, 1, -1
! 250: IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
! 251: $ RETURN
! 252: END DO
! 253: ELSE
! 254: *
! 255: * Lower triangular storage: examine D from top to bottom.
! 256: *
! 257: DO I = 1, N
! 258: IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO )
! 259: $ RETURN
! 260: END DO
! 261: END IF
! 262: *
! 263: * Estimate the 1-norm of the inverse.
! 264: *
! 265: KASE = 0
! 266: 30 CONTINUE
! 267: CALL ZLACN2( N, WORK( N+1 ), WORK, AINVNM, KASE, ISAVE )
! 268: IF( KASE.NE.0 ) THEN
! 269: *
! 270: * Multiply by inv(L*D*L**H) or inv(U*D*U**H).
! 271: *
! 272: CALL ZHETRS_3( UPLO, N, 1, A, LDA, E, IPIV, WORK, N, INFO )
! 273: GO TO 30
! 274: END IF
! 275: *
! 276: * Compute the estimate of the reciprocal condition number.
! 277: *
! 278: IF( AINVNM.NE.ZERO )
! 279: $ RCOND = ( ONE / AINVNM ) / ANORM
! 280: *
! 281: RETURN
! 282: *
! 283: * End of ZHECON_3
! 284: *
! 285: END
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