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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE ZLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, 00002 $ VN2, AUXV, F, LDF ) 00003 * 00004 * -- LAPACK auxiliary routine (version 3.3.1) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * -- April 2011 -- 00008 * 00009 * .. Scalar Arguments .. 00010 INTEGER KB, LDA, LDF, M, N, NB, OFFSET 00011 * .. 00012 * .. Array Arguments .. 00013 INTEGER JPVT( * ) 00014 DOUBLE PRECISION VN1( * ), VN2( * ) 00015 COMPLEX*16 A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ) 00016 * .. 00017 * 00018 * Purpose 00019 * ======= 00020 * 00021 * ZLAQPS computes a step of QR factorization with column pivoting 00022 * of a complex M-by-N matrix A by using Blas-3. It tries to factorize 00023 * NB columns from A starting from the row OFFSET+1, and updates all 00024 * of the matrix with Blas-3 xGEMM. 00025 * 00026 * In some cases, due to catastrophic cancellations, it cannot 00027 * factorize NB columns. Hence, the actual number of factorized 00028 * columns is returned in KB. 00029 * 00030 * Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. 00031 * 00032 * Arguments 00033 * ========= 00034 * 00035 * M (input) INTEGER 00036 * The number of rows of the matrix A. M >= 0. 00037 * 00038 * N (input) INTEGER 00039 * The number of columns of the matrix A. N >= 0 00040 * 00041 * OFFSET (input) INTEGER 00042 * The number of rows of A that have been factorized in 00043 * previous steps. 00044 * 00045 * NB (input) INTEGER 00046 * The number of columns to factorize. 00047 * 00048 * KB (output) INTEGER 00049 * The number of columns actually factorized. 00050 * 00051 * A (input/output) COMPLEX*16 array, dimension (LDA,N) 00052 * On entry, the M-by-N matrix A. 00053 * On exit, block A(OFFSET+1:M,1:KB) is the triangular 00054 * factor obtained and block A(1:OFFSET,1:N) has been 00055 * accordingly pivoted, but no factorized. 00056 * The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has 00057 * been updated. 00058 * 00059 * LDA (input) INTEGER 00060 * The leading dimension of the array A. LDA >= max(1,M). 00061 * 00062 * JPVT (input/output) INTEGER array, dimension (N) 00063 * JPVT(I) = K <==> Column K of the full matrix A has been 00064 * permuted into position I in AP. 00065 * 00066 * TAU (output) COMPLEX*16 array, dimension (KB) 00067 * The scalar factors of the elementary reflectors. 00068 * 00069 * VN1 (input/output) DOUBLE PRECISION array, dimension (N) 00070 * The vector with the partial column norms. 00071 * 00072 * VN2 (input/output) DOUBLE PRECISION array, dimension (N) 00073 * The vector with the exact column norms. 00074 * 00075 * AUXV (input/output) COMPLEX*16 array, dimension (NB) 00076 * Auxiliar vector. 00077 * 00078 * F (input/output) COMPLEX*16 array, dimension (LDF,NB) 00079 * Matrix F**H = L * Y**H * A. 00080 * 00081 * LDF (input) INTEGER 00082 * The leading dimension of the array F. LDF >= max(1,N). 00083 * 00084 * Further Details 00085 * =============== 00086 * 00087 * Based on contributions by 00088 * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain 00089 * X. Sun, Computer Science Dept., Duke University, USA 00090 * 00091 * Partial column norm updating strategy modified by 00092 * Z. Drmac and Z. Bujanovic, Dept. of Mathematics, 00093 * University of Zagreb, Croatia. 00094 * -- April 2011 -- 00095 * For more details see LAPACK Working Note 176. 00096 * ===================================================================== 00097 * 00098 * .. Parameters .. 00099 DOUBLE PRECISION ZERO, ONE 00100 COMPLEX*16 CZERO, CONE 00101 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, 00102 $ CZERO = ( 0.0D+0, 0.0D+0 ), 00103 $ CONE = ( 1.0D+0, 0.0D+0 ) ) 00104 * .. 00105 * .. Local Scalars .. 00106 INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK 00107 DOUBLE PRECISION TEMP, TEMP2, TOL3Z 00108 COMPLEX*16 AKK 00109 * .. 00110 * .. External Subroutines .. 00111 EXTERNAL ZGEMM, ZGEMV, ZLARFG, ZSWAP 00112 * .. 00113 * .. Intrinsic Functions .. 00114 INTRINSIC ABS, DBLE, DCONJG, MAX, MIN, NINT, SQRT 00115 * .. 00116 * .. External Functions .. 00117 INTEGER IDAMAX 00118 DOUBLE PRECISION DLAMCH, DZNRM2 00119 EXTERNAL IDAMAX, DLAMCH, DZNRM2 00120 * .. 00121 * .. Executable Statements .. 00122 * 00123 LASTRK = MIN( M, N+OFFSET ) 00124 LSTICC = 0 00125 K = 0 00126 TOL3Z = SQRT(DLAMCH('Epsilon')) 00127 * 00128 * Beginning of while loop. 00129 * 00130 10 CONTINUE 00131 IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN 00132 K = K + 1 00133 RK = OFFSET + K 00134 * 00135 * Determine ith pivot column and swap if necessary 00136 * 00137 PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 ) 00138 IF( PVT.NE.K ) THEN 00139 CALL ZSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 ) 00140 CALL ZSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF ) 00141 ITEMP = JPVT( PVT ) 00142 JPVT( PVT ) = JPVT( K ) 00143 JPVT( K ) = ITEMP 00144 VN1( PVT ) = VN1( K ) 00145 VN2( PVT ) = VN2( K ) 00146 END IF 00147 * 00148 * Apply previous Householder reflectors to column K: 00149 * A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)**H. 00150 * 00151 IF( K.GT.1 ) THEN 00152 DO 20 J = 1, K - 1 00153 F( K, J ) = DCONJG( F( K, J ) ) 00154 20 CONTINUE 00155 CALL ZGEMV( 'No transpose', M-RK+1, K-1, -CONE, A( RK, 1 ), 00156 $ LDA, F( K, 1 ), LDF, CONE, A( RK, K ), 1 ) 00157 DO 30 J = 1, K - 1 00158 F( K, J ) = DCONJG( F( K, J ) ) 00159 30 CONTINUE 00160 END IF 00161 * 00162 * Generate elementary reflector H(k). 00163 * 00164 IF( RK.LT.M ) THEN 00165 CALL ZLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) ) 00166 ELSE 00167 CALL ZLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) ) 00168 END IF 00169 * 00170 AKK = A( RK, K ) 00171 A( RK, K ) = CONE 00172 * 00173 * Compute Kth column of F: 00174 * 00175 * Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)**H*A(RK:M,K). 00176 * 00177 IF( K.LT.N ) THEN 00178 CALL ZGEMV( 'Conjugate transpose', M-RK+1, N-K, TAU( K ), 00179 $ A( RK, K+1 ), LDA, A( RK, K ), 1, CZERO, 00180 $ F( K+1, K ), 1 ) 00181 END IF 00182 * 00183 * Padding F(1:K,K) with zeros. 00184 * 00185 DO 40 J = 1, K 00186 F( J, K ) = CZERO 00187 40 CONTINUE 00188 * 00189 * Incremental updating of F: 00190 * F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)**H 00191 * *A(RK:M,K). 00192 * 00193 IF( K.GT.1 ) THEN 00194 CALL ZGEMV( 'Conjugate transpose', M-RK+1, K-1, -TAU( K ), 00195 $ A( RK, 1 ), LDA, A( RK, K ), 1, CZERO, 00196 $ AUXV( 1 ), 1 ) 00197 * 00198 CALL ZGEMV( 'No transpose', N, K-1, CONE, F( 1, 1 ), LDF, 00199 $ AUXV( 1 ), 1, CONE, F( 1, K ), 1 ) 00200 END IF 00201 * 00202 * Update the current row of A: 00203 * A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)**H. 00204 * 00205 IF( K.LT.N ) THEN 00206 CALL ZGEMM( 'No transpose', 'Conjugate transpose', 1, N-K, 00207 $ K, -CONE, A( RK, 1 ), LDA, F( K+1, 1 ), LDF, 00208 $ CONE, A( RK, K+1 ), LDA ) 00209 END IF 00210 * 00211 * Update partial column norms. 00212 * 00213 IF( RK.LT.LASTRK ) THEN 00214 DO 50 J = K + 1, N 00215 IF( VN1( J ).NE.ZERO ) THEN 00216 * 00217 * NOTE: The following 4 lines follow from the analysis in 00218 * Lapack Working Note 176. 00219 * 00220 TEMP = ABS( A( RK, J ) ) / VN1( J ) 00221 TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) 00222 TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2 00223 IF( TEMP2 .LE. TOL3Z ) THEN 00224 VN2( J ) = DBLE( LSTICC ) 00225 LSTICC = J 00226 ELSE 00227 VN1( J ) = VN1( J )*SQRT( TEMP ) 00228 END IF 00229 END IF 00230 50 CONTINUE 00231 END IF 00232 * 00233 A( RK, K ) = AKK 00234 * 00235 * End of while loop. 00236 * 00237 GO TO 10 00238 END IF 00239 KB = K 00240 RK = OFFSET + KB 00241 * 00242 * Apply the block reflector to the rest of the matrix: 00243 * A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) - 00244 * A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)**H. 00245 * 00246 IF( KB.LT.MIN( N, M-OFFSET ) ) THEN 00247 CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-RK, N-KB, 00248 $ KB, -CONE, A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, 00249 $ CONE, A( RK+1, KB+1 ), LDA ) 00250 END IF 00251 * 00252 * Recomputation of difficult columns. 00253 * 00254 60 CONTINUE 00255 IF( LSTICC.GT.0 ) THEN 00256 ITEMP = NINT( VN2( LSTICC ) ) 00257 VN1( LSTICC ) = DZNRM2( M-RK, A( RK+1, LSTICC ), 1 ) 00258 * 00259 * NOTE: The computation of VN1( LSTICC ) relies on the fact that 00260 * SNRM2 does not fail on vectors with norm below the value of 00261 * SQRT(DLAMCH('S')) 00262 * 00263 VN2( LSTICC ) = VN1( LSTICC ) 00264 LSTICC = ITEMP 00265 GO TO 60 00266 END IF 00267 * 00268 RETURN 00269 * 00270 * End of ZLAQPS 00271 * 00272 END
1.7.3 
