Annotation of rpl/lapack/lapack/zpbstf.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZPBSTF
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZPBSTF + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpbstf.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpbstf.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpbstf.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * CHARACTER UPLO
! 25: * INTEGER INFO, KD, LDAB, N
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * COMPLEX*16 AB( LDAB, * )
! 29: * ..
! 30: *
! 31: *
! 32: *> \par Purpose:
! 33: * =============
! 34: *>
! 35: *> \verbatim
! 36: *>
! 37: *> ZPBSTF computes a split Cholesky factorization of a complex
! 38: *> Hermitian positive definite band matrix A.
! 39: *>
! 40: *> This routine is designed to be used in conjunction with ZHBGST.
! 41: *>
! 42: *> The factorization has the form A = S**H*S where S is a band matrix
! 43: *> of the same bandwidth as A and the following structure:
! 44: *>
! 45: *> S = ( U )
! 46: *> ( M L )
! 47: *>
! 48: *> where U is upper triangular of order m = (n+kd)/2, and L is lower
! 49: *> triangular of order n-m.
! 50: *> \endverbatim
! 51: *
! 52: * Arguments:
! 53: * ==========
! 54: *
! 55: *> \param[in] UPLO
! 56: *> \verbatim
! 57: *> UPLO is CHARACTER*1
! 58: *> = 'U': Upper triangle of A is stored;
! 59: *> = 'L': Lower triangle of A is stored.
! 60: *> \endverbatim
! 61: *>
! 62: *> \param[in] N
! 63: *> \verbatim
! 64: *> N is INTEGER
! 65: *> The order of the matrix A. N >= 0.
! 66: *> \endverbatim
! 67: *>
! 68: *> \param[in] KD
! 69: *> \verbatim
! 70: *> KD is INTEGER
! 71: *> The number of superdiagonals of the matrix A if UPLO = 'U',
! 72: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 73: *> \endverbatim
! 74: *>
! 75: *> \param[in,out] AB
! 76: *> \verbatim
! 77: *> AB is COMPLEX*16 array, dimension (LDAB,N)
! 78: *> On entry, the upper or lower triangle of the Hermitian band
! 79: *> matrix A, stored in the first kd+1 rows of the array. The
! 80: *> j-th column of A is stored in the j-th column of the array AB
! 81: *> as follows:
! 82: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 83: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 84: *>
! 85: *> On exit, if INFO = 0, the factor S from the split Cholesky
! 86: *> factorization A = S**H*S. See Further Details.
! 87: *> \endverbatim
! 88: *>
! 89: *> \param[in] LDAB
! 90: *> \verbatim
! 91: *> LDAB is INTEGER
! 92: *> The leading dimension of the array AB. LDAB >= KD+1.
! 93: *> \endverbatim
! 94: *>
! 95: *> \param[out] INFO
! 96: *> \verbatim
! 97: *> INFO is INTEGER
! 98: *> = 0: successful exit
! 99: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 100: *> > 0: if INFO = i, the factorization could not be completed,
! 101: *> because the updated element a(i,i) was negative; the
! 102: *> matrix A is not positive definite.
! 103: *> \endverbatim
! 104: *
! 105: * Authors:
! 106: * ========
! 107: *
! 108: *> \author Univ. of Tennessee
! 109: *> \author Univ. of California Berkeley
! 110: *> \author Univ. of Colorado Denver
! 111: *> \author NAG Ltd.
! 112: *
! 113: *> \date November 2011
! 114: *
! 115: *> \ingroup complex16OTHERcomputational
! 116: *
! 117: *> \par Further Details:
! 118: * =====================
! 119: *>
! 120: *> \verbatim
! 121: *>
! 122: *> The band storage scheme is illustrated by the following example, when
! 123: *> N = 7, KD = 2:
! 124: *>
! 125: *> S = ( s11 s12 s13 )
! 126: *> ( s22 s23 s24 )
! 127: *> ( s33 s34 )
! 128: *> ( s44 )
! 129: *> ( s53 s54 s55 )
! 130: *> ( s64 s65 s66 )
! 131: *> ( s75 s76 s77 )
! 132: *>
! 133: *> If UPLO = 'U', the array AB holds:
! 134: *>
! 135: *> on entry: on exit:
! 136: *>
! 137: *> * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H
! 138: *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
! 139: *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
! 140: *>
! 141: *> If UPLO = 'L', the array AB holds:
! 142: *>
! 143: *> on entry: on exit:
! 144: *>
! 145: *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
! 146: *> a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 *
! 147: *> a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
! 148: *>
! 149: *> Array elements marked * are not used by the routine; s12**H denotes
! 150: *> conjg(s12); the diagonal elements of S are real.
! 151: *> \endverbatim
! 152: *>
! 153: * =====================================================================
1.1 bertrand 154: SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
155: *
1.9 ! bertrand 156: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 157: * -- LAPACK is a software package provided by Univ. of Tennessee, --
158: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 159: * November 2011
1.1 bertrand 160: *
161: * .. Scalar Arguments ..
162: CHARACTER UPLO
163: INTEGER INFO, KD, LDAB, N
164: * ..
165: * .. Array Arguments ..
166: COMPLEX*16 AB( LDAB, * )
167: * ..
168: *
169: * =====================================================================
170: *
171: * .. Parameters ..
172: DOUBLE PRECISION ONE, ZERO
173: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
174: * ..
175: * .. Local Scalars ..
176: LOGICAL UPPER
177: INTEGER J, KLD, KM, M
178: DOUBLE PRECISION AJJ
179: * ..
180: * .. External Functions ..
181: LOGICAL LSAME
182: EXTERNAL LSAME
183: * ..
184: * .. External Subroutines ..
185: EXTERNAL XERBLA, ZDSCAL, ZHER, ZLACGV
186: * ..
187: * .. Intrinsic Functions ..
188: INTRINSIC DBLE, MAX, MIN, SQRT
189: * ..
190: * .. Executable Statements ..
191: *
192: * Test the input parameters.
193: *
194: INFO = 0
195: UPPER = LSAME( UPLO, 'U' )
196: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
197: INFO = -1
198: ELSE IF( N.LT.0 ) THEN
199: INFO = -2
200: ELSE IF( KD.LT.0 ) THEN
201: INFO = -3
202: ELSE IF( LDAB.LT.KD+1 ) THEN
203: INFO = -5
204: END IF
205: IF( INFO.NE.0 ) THEN
206: CALL XERBLA( 'ZPBSTF', -INFO )
207: RETURN
208: END IF
209: *
210: * Quick return if possible
211: *
212: IF( N.EQ.0 )
213: $ RETURN
214: *
215: KLD = MAX( 1, LDAB-1 )
216: *
217: * Set the splitting point m.
218: *
219: M = ( N+KD ) / 2
220: *
221: IF( UPPER ) THEN
222: *
223: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
224: *
225: DO 10 J = N, M + 1, -1
226: *
227: * Compute s(j,j) and test for non-positive-definiteness.
228: *
229: AJJ = DBLE( AB( KD+1, J ) )
230: IF( AJJ.LE.ZERO ) THEN
231: AB( KD+1, J ) = AJJ
232: GO TO 50
233: END IF
234: AJJ = SQRT( AJJ )
235: AB( KD+1, J ) = AJJ
236: KM = MIN( J-1, KD )
237: *
238: * Compute elements j-km:j-1 of the j-th column and update the
239: * the leading submatrix within the band.
240: *
241: CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
242: CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
243: $ AB( KD+1, J-KM ), KLD )
244: 10 CONTINUE
245: *
246: * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
247: *
248: DO 20 J = 1, M
249: *
250: * Compute s(j,j) and test for non-positive-definiteness.
251: *
252: AJJ = DBLE( AB( KD+1, J ) )
253: IF( AJJ.LE.ZERO ) THEN
254: AB( KD+1, J ) = AJJ
255: GO TO 50
256: END IF
257: AJJ = SQRT( AJJ )
258: AB( KD+1, J ) = AJJ
259: KM = MIN( KD, M-J )
260: *
261: * Compute elements j+1:j+km of the j-th row and update the
262: * trailing submatrix within the band.
263: *
264: IF( KM.GT.0 ) THEN
265: CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
266: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
267: CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
268: $ AB( KD+1, J+1 ), KLD )
269: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
270: END IF
271: 20 CONTINUE
272: ELSE
273: *
274: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
275: *
276: DO 30 J = N, M + 1, -1
277: *
278: * Compute s(j,j) and test for non-positive-definiteness.
279: *
280: AJJ = DBLE( AB( 1, J ) )
281: IF( AJJ.LE.ZERO ) THEN
282: AB( 1, J ) = AJJ
283: GO TO 50
284: END IF
285: AJJ = SQRT( AJJ )
286: AB( 1, J ) = AJJ
287: KM = MIN( J-1, KD )
288: *
289: * Compute elements j-km:j-1 of the j-th row and update the
290: * trailing submatrix within the band.
291: *
292: CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
293: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
294: CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
295: $ AB( 1, J-KM ), KLD )
296: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
297: 30 CONTINUE
298: *
299: * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
300: *
301: DO 40 J = 1, M
302: *
303: * Compute s(j,j) and test for non-positive-definiteness.
304: *
305: AJJ = DBLE( AB( 1, J ) )
306: IF( AJJ.LE.ZERO ) THEN
307: AB( 1, J ) = AJJ
308: GO TO 50
309: END IF
310: AJJ = SQRT( AJJ )
311: AB( 1, J ) = AJJ
312: KM = MIN( KD, M-J )
313: *
314: * Compute elements j+1:j+km of the j-th column and update the
315: * trailing submatrix within the band.
316: *
317: IF( KM.GT.0 ) THEN
318: CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
319: CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1,
320: $ AB( 1, J+1 ), KLD )
321: END IF
322: 40 CONTINUE
323: END IF
324: RETURN
325: *
326: 50 CONTINUE
327: INFO = J
328: RETURN
329: *
330: * End of ZPBSTF
331: *
332: END
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