00001 SUBROUTINE CGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, 00002 $ INFO ) 00003 * 00004 * -- LAPACK routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * .. Scalar Arguments .. 00010 INTEGER INFO, LDA, LWORK, M, N 00011 * .. 00012 * .. Array Arguments .. 00013 REAL D( * ), E( * ) 00014 COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), 00015 $ WORK( * ) 00016 * .. 00017 * 00018 * Purpose 00019 * ======= 00020 * 00021 * CGEBRD reduces a general complex M-by-N matrix A to upper or lower 00022 * bidiagonal form B by a unitary transformation: Q**H * A * P = B. 00023 * 00024 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. 00025 * 00026 * Arguments 00027 * ========= 00028 * 00029 * M (input) INTEGER 00030 * The number of rows in the matrix A. M >= 0. 00031 * 00032 * N (input) INTEGER 00033 * The number of columns in the matrix A. N >= 0. 00034 * 00035 * A (input/output) COMPLEX array, dimension (LDA,N) 00036 * On entry, the M-by-N general matrix to be reduced. 00037 * On exit, 00038 * if m >= n, the diagonal and the first superdiagonal are 00039 * overwritten with the upper bidiagonal matrix B; the 00040 * elements below the diagonal, with the array TAUQ, represent 00041 * the unitary matrix Q as a product of elementary 00042 * reflectors, and the elements above the first superdiagonal, 00043 * with the array TAUP, represent the unitary matrix P as 00044 * a product of elementary reflectors; 00045 * if m < n, the diagonal and the first subdiagonal are 00046 * overwritten with the lower bidiagonal matrix B; the 00047 * elements below the first subdiagonal, with the array TAUQ, 00048 * represent the unitary matrix Q as a product of 00049 * elementary reflectors, and the elements above the diagonal, 00050 * with the array TAUP, represent the unitary matrix P as 00051 * a product of elementary reflectors. 00052 * See Further Details. 00053 * 00054 * LDA (input) INTEGER 00055 * The leading dimension of the array A. LDA >= max(1,M). 00056 * 00057 * D (output) REAL array, dimension (min(M,N)) 00058 * The diagonal elements of the bidiagonal matrix B: 00059 * D(i) = A(i,i). 00060 * 00061 * E (output) REAL array, dimension (min(M,N)-1) 00062 * The off-diagonal elements of the bidiagonal matrix B: 00063 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; 00064 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 00065 * 00066 * TAUQ (output) COMPLEX array dimension (min(M,N)) 00067 * The scalar factors of the elementary reflectors which 00068 * represent the unitary matrix Q. See Further Details. 00069 * 00070 * TAUP (output) COMPLEX array, dimension (min(M,N)) 00071 * The scalar factors of the elementary reflectors which 00072 * represent the unitary matrix P. See Further Details. 00073 * 00074 * WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) 00075 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00076 * 00077 * LWORK (input) INTEGER 00078 * The length of the array WORK. LWORK >= max(1,M,N). 00079 * For optimum performance LWORK >= (M+N)*NB, where NB 00080 * is the optimal blocksize. 00081 * 00082 * If LWORK = -1, then a workspace query is assumed; the routine 00083 * only calculates the optimal size of the WORK array, returns 00084 * this value as the first entry of the WORK array, and no error 00085 * message related to LWORK is issued by XERBLA. 00086 * 00087 * INFO (output) INTEGER 00088 * = 0: successful exit. 00089 * < 0: if INFO = -i, the i-th argument had an illegal value. 00090 * 00091 * Further Details 00092 * =============== 00093 * 00094 * The matrices Q and P are represented as products of elementary 00095 * reflectors: 00096 * 00097 * If m >= n, 00098 * 00099 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 00100 * 00101 * Each H(i) and G(i) has the form: 00102 * 00103 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 00104 * 00105 * where tauq and taup are complex scalars, and v and u are complex 00106 * vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in 00107 * A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in 00108 * A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00109 * 00110 * If m < n, 00111 * 00112 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 00113 * 00114 * Each H(i) and G(i) has the form: 00115 * 00116 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 00117 * 00118 * where tauq and taup are complex scalars, and v and u are complex 00119 * vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in 00120 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in 00121 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00122 * 00123 * The contents of A on exit are illustrated by the following examples: 00124 * 00125 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 00126 * 00127 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 00128 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 00129 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 00130 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 00131 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 00132 * ( v1 v2 v3 v4 v5 ) 00133 * 00134 * where d and e denote diagonal and off-diagonal elements of B, vi 00135 * denotes an element of the vector defining H(i), and ui an element of 00136 * the vector defining G(i). 00137 * 00138 * ===================================================================== 00139 * 00140 * .. Parameters .. 00141 COMPLEX ONE 00142 PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ) ) 00143 * .. 00144 * .. Local Scalars .. 00145 LOGICAL LQUERY 00146 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB, 00147 $ NBMIN, NX 00148 REAL WS 00149 * .. 00150 * .. External Subroutines .. 00151 EXTERNAL CGEBD2, CGEMM, CLABRD, XERBLA 00152 * .. 00153 * .. Intrinsic Functions .. 00154 INTRINSIC MAX, MIN, REAL 00155 * .. 00156 * .. External Functions .. 00157 INTEGER ILAENV 00158 EXTERNAL ILAENV 00159 * .. 00160 * .. Executable Statements .. 00161 * 00162 * Test the input parameters 00163 * 00164 INFO = 0 00165 NB = MAX( 1, ILAENV( 1, 'CGEBRD', ' ', M, N, -1, -1 ) ) 00166 LWKOPT = ( M+N )*NB 00167 WORK( 1 ) = REAL( LWKOPT ) 00168 LQUERY = ( LWORK.EQ.-1 ) 00169 IF( M.LT.0 ) THEN 00170 INFO = -1 00171 ELSE IF( N.LT.0 ) THEN 00172 INFO = -2 00173 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00174 INFO = -4 00175 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN 00176 INFO = -10 00177 END IF 00178 IF( INFO.LT.0 ) THEN 00179 CALL XERBLA( 'CGEBRD', -INFO ) 00180 RETURN 00181 ELSE IF( LQUERY ) THEN 00182 RETURN 00183 END IF 00184 * 00185 * Quick return if possible 00186 * 00187 MINMN = MIN( M, N ) 00188 IF( MINMN.EQ.0 ) THEN 00189 WORK( 1 ) = 1 00190 RETURN 00191 END IF 00192 * 00193 WS = MAX( M, N ) 00194 LDWRKX = M 00195 LDWRKY = N 00196 * 00197 IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN 00198 * 00199 * Set the crossover point NX. 00200 * 00201 NX = MAX( NB, ILAENV( 3, 'CGEBRD', ' ', M, N, -1, -1 ) ) 00202 * 00203 * Determine when to switch from blocked to unblocked code. 00204 * 00205 IF( NX.LT.MINMN ) THEN 00206 WS = ( M+N )*NB 00207 IF( LWORK.LT.WS ) THEN 00208 * 00209 * Not enough work space for the optimal NB, consider using 00210 * a smaller block size. 00211 * 00212 NBMIN = ILAENV( 2, 'CGEBRD', ' ', M, N, -1, -1 ) 00213 IF( LWORK.GE.( M+N )*NBMIN ) THEN 00214 NB = LWORK / ( M+N ) 00215 ELSE 00216 NB = 1 00217 NX = MINMN 00218 END IF 00219 END IF 00220 END IF 00221 ELSE 00222 NX = MINMN 00223 END IF 00224 * 00225 DO 30 I = 1, MINMN - NX, NB 00226 * 00227 * Reduce rows and columns i:i+ib-1 to bidiagonal form and return 00228 * the matrices X and Y which are needed to update the unreduced 00229 * part of the matrix 00230 * 00231 CALL CLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ), 00232 $ TAUQ( I ), TAUP( I ), WORK, LDWRKX, 00233 $ WORK( LDWRKX*NB+1 ), LDWRKY ) 00234 * 00235 * Update the trailing submatrix A(i+ib:m,i+ib:n), using 00236 * an update of the form A := A - V*Y' - X*U' 00237 * 00238 CALL CGEMM( 'No transpose', 'Conjugate transpose', M-I-NB+1, 00239 $ N-I-NB+1, NB, -ONE, A( I+NB, I ), LDA, 00240 $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE, 00241 $ A( I+NB, I+NB ), LDA ) 00242 CALL CGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1, 00243 $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA, 00244 $ ONE, A( I+NB, I+NB ), LDA ) 00245 * 00246 * Copy diagonal and off-diagonal elements of B back into A 00247 * 00248 IF( M.GE.N ) THEN 00249 DO 10 J = I, I + NB - 1 00250 A( J, J ) = D( J ) 00251 A( J, J+1 ) = E( J ) 00252 10 CONTINUE 00253 ELSE 00254 DO 20 J = I, I + NB - 1 00255 A( J, J ) = D( J ) 00256 A( J+1, J ) = E( J ) 00257 20 CONTINUE 00258 END IF 00259 30 CONTINUE 00260 * 00261 * Use unblocked code to reduce the remainder of the matrix 00262 * 00263 CALL CGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ), 00264 $ TAUQ( I ), TAUP( I ), WORK, IINFO ) 00265 WORK( 1 ) = WS 00266 RETURN 00267 * 00268 * End of CGEBRD 00269 * 00270 END
1.7.2