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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) 2: * 3: * -- LAPACK auxiliary routine (version 3.2) -- 4: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 6: * November 2006 7: * 8: * .. Scalar Arguments .. 9: INTEGER K, LDA, LDT, LDY, N, NB 10: * .. 11: * .. Array Arguments .. 12: COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ), 13: $ Y( LDY, NB ) 14: * .. 15: * 16: * Purpose 17: * ======= 18: * 19: * ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1) 20: * matrix A so that elements below the k-th subdiagonal are zero. The 21: * reduction is performed by a unitary similarity transformation 22: * Q' * A * Q. The routine returns the matrices V and T which determine 23: * Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. 24: * 25: * This is an OBSOLETE auxiliary routine. 26: * This routine will be 'deprecated' in a future release. 27: * Please use the new routine ZLAHR2 instead. 28: * 29: * Arguments 30: * ========= 31: * 32: * N (input) INTEGER 33: * The order of the matrix A. 34: * 35: * K (input) INTEGER 36: * The offset for the reduction. Elements below the k-th 37: * subdiagonal in the first NB columns are reduced to zero. 38: * 39: * NB (input) INTEGER 40: * The number of columns to be reduced. 41: * 42: * A (input/output) COMPLEX*16 array, dimension (LDA,N-K+1) 43: * On entry, the n-by-(n-k+1) general matrix A. 44: * On exit, the elements on and above the k-th subdiagonal in 45: * the first NB columns are overwritten with the corresponding 46: * elements of the reduced matrix; the elements below the k-th 47: * subdiagonal, with the array TAU, represent the matrix Q as a 48: * product of elementary reflectors. The other columns of A are 49: * unchanged. See Further Details. 50: * 51: * LDA (input) INTEGER 52: * The leading dimension of the array A. LDA >= max(1,N). 53: * 54: * TAU (output) COMPLEX*16 array, dimension (NB) 55: * The scalar factors of the elementary reflectors. See Further 56: * Details. 57: * 58: * T (output) COMPLEX*16 array, dimension (LDT,NB) 59: * The upper triangular matrix T. 60: * 61: * LDT (input) INTEGER 62: * The leading dimension of the array T. LDT >= NB. 63: * 64: * Y (output) COMPLEX*16 array, dimension (LDY,NB) 65: * The n-by-nb matrix Y. 66: * 67: * LDY (input) INTEGER 68: * The leading dimension of the array Y. LDY >= max(1,N). 69: * 70: * Further Details 71: * =============== 72: * 73: * The matrix Q is represented as a product of nb elementary reflectors 74: * 75: * Q = H(1) H(2) . . . H(nb). 76: * 77: * Each H(i) has the form 78: * 79: * H(i) = I - tau * v * v' 80: * 81: * where tau is a complex scalar, and v is a complex vector with 82: * v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in 83: * A(i+k+1:n,i), and tau in TAU(i). 84: * 85: * The elements of the vectors v together form the (n-k+1)-by-nb matrix 86: * V which is needed, with T and Y, to apply the transformation to the 87: * unreduced part of the matrix, using an update of the form: 88: * A := (I - V*T*V') * (A - Y*V'). 89: * 90: * The contents of A on exit are illustrated by the following example 91: * with n = 7, k = 3 and nb = 2: 92: * 93: * ( a h a a a ) 94: * ( a h a a a ) 95: * ( a h a a a ) 96: * ( h h a a a ) 97: * ( v1 h a a a ) 98: * ( v1 v2 a a a ) 99: * ( v1 v2 a a a ) 100: * 101: * where a denotes an element of the original matrix A, h denotes a 102: * modified element of the upper Hessenberg matrix H, and vi denotes an 103: * element of the vector defining H(i). 104: * 105: * ===================================================================== 106: * 107: * .. Parameters .. 108: COMPLEX*16 ZERO, ONE 109: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), 110: $ ONE = ( 1.0D+0, 0.0D+0 ) ) 111: * .. 112: * .. Local Scalars .. 113: INTEGER I 114: COMPLEX*16 EI 115: * .. 116: * .. External Subroutines .. 117: EXTERNAL ZAXPY, ZCOPY, ZGEMV, ZLACGV, ZLARFG, ZSCAL, 118: $ ZTRMV 119: * .. 120: * .. Intrinsic Functions .. 121: INTRINSIC MIN 122: * .. 123: * .. Executable Statements .. 124: * 125: * Quick return if possible 126: * 127: IF( N.LE.1 ) 128: $ RETURN 129: * 130: DO 10 I = 1, NB 131: IF( I.GT.1 ) THEN 132: * 133: * Update A(1:n,i) 134: * 135: * Compute i-th column of A - Y * V' 136: * 137: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA ) 138: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, 139: $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) 140: CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA ) 141: * 142: * Apply I - V * T' * V' to this column (call it b) from the 143: * left, using the last column of T as workspace 144: * 145: * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) 146: * ( V2 ) ( b2 ) 147: * 148: * where V1 is unit lower triangular 149: * 150: * w := V1' * b1 151: * 152: CALL ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) 153: CALL ZTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1, 154: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) 155: * 156: * w := w + V2'*b2 157: * 158: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE, 159: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ONE, 160: $ T( 1, NB ), 1 ) 161: * 162: * w := T'*w 163: * 164: CALL ZTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1, 165: $ T, LDT, T( 1, NB ), 1 ) 166: * 167: * b2 := b2 - V2*w 168: * 169: CALL ZGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ), 170: $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) 171: * 172: * b1 := b1 - V1*w 173: * 174: CALL ZTRMV( 'Lower', 'No transpose', 'Unit', I-1, 175: $ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) 176: CALL ZAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) 177: * 178: A( K+I-1, I-1 ) = EI 179: END IF 180: * 181: * Generate the elementary reflector H(i) to annihilate 182: * A(k+i+1:n,i) 183: * 184: EI = A( K+I, I ) 185: CALL ZLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1, 186: $ TAU( I ) ) 187: A( K+I, I ) = ONE 188: * 189: * Compute Y(1:n,i) 190: * 191: CALL ZGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, 192: $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) 193: CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE, 194: $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ), 195: $ 1 ) 196: CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, 197: $ ONE, Y( 1, I ), 1 ) 198: CALL ZSCAL( N, TAU( I ), Y( 1, I ), 1 ) 199: * 200: * Compute T(1:i,i) 201: * 202: CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) 203: CALL ZTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT, 204: $ T( 1, I ), 1 ) 205: T( I, I ) = TAU( I ) 206: * 207: 10 CONTINUE 208: A( K+NB, NB ) = EI 209: * 210: RETURN 211: * 212: * End of ZLAHRD 213: * 214: END