10#ifndef EIGEN_CXX11_TENSOR_TENSOR_FFT_H
11#define EIGEN_CXX11_TENSOR_TENSOR_FFT_H
26template <
bool NeedUprade>
struct MakeComplex {
29 T operator() (
const T& val)
const {
return val; }
32template <>
struct MakeComplex<true> {
35 std::complex<T> operator() (
const T& val)
const {
return std::complex<T>(val, 0); }
38template <>
struct MakeComplex<false> {
41 std::complex<T> operator() (
const std::complex<T>& val)
const {
return val; }
44template <
int ResultType>
struct PartOf {
45 template <
typename T> T operator() (
const T& val)
const {
return val; }
48template <>
struct PartOf<RealPart> {
49 template <
typename T> T operator() (
const std::complex<T>& val)
const {
return val.real(); }
52template <>
struct PartOf<ImagPart> {
53 template <
typename T> T operator() (
const std::complex<T>& val)
const {
return val.imag(); }
57template <
typename FFT,
typename XprType,
int FFTResultType,
int FFTDir>
58struct traits<TensorFFTOp<FFT, XprType, FFTResultType, FFTDir> > :
public traits<XprType> {
59 typedef traits<XprType> XprTraits;
60 typedef typename NumTraits<typename XprTraits::Scalar>::Real RealScalar;
61 typedef typename std::complex<RealScalar> ComplexScalar;
62 typedef typename XprTraits::Scalar InputScalar;
63 typedef typename conditional<FFTResultType == RealPart || FFTResultType == ImagPart, RealScalar, ComplexScalar>::type OutputScalar;
64 typedef typename XprTraits::StorageKind StorageKind;
65 typedef typename XprTraits::Index
Index;
66 typedef typename XprType::Nested Nested;
67 typedef typename remove_reference<Nested>::type _Nested;
68 static const int NumDimensions = XprTraits::NumDimensions;
69 static const int Layout = XprTraits::Layout;
70 typedef typename traits<XprType>::PointerType PointerType;
73template <
typename FFT,
typename XprType,
int FFTResultType,
int FFTDirection>
74struct eval<TensorFFTOp<FFT, XprType, FFTResultType, FFTDirection>,
Eigen::Dense> {
75 typedef const TensorFFTOp<FFT, XprType, FFTResultType, FFTDirection>& type;
78template <
typename FFT,
typename XprType,
int FFTResultType,
int FFTDirection>
79struct nested<TensorFFTOp<FFT, XprType, FFTResultType, FFTDirection>, 1, typename eval<TensorFFTOp<FFT, XprType, FFTResultType, FFTDirection> >::type> {
80 typedef TensorFFTOp<FFT, XprType, FFTResultType, FFTDirection> type;
85template <
typename FFT,
typename XprType,
int FFTResultType,
int FFTDir>
86class TensorFFTOp :
public TensorBase<TensorFFTOp<FFT, XprType, FFTResultType, FFTDir>, ReadOnlyAccessors> {
88 typedef typename Eigen::internal::traits<TensorFFTOp>::Scalar Scalar;
90 typedef typename std::complex<RealScalar> ComplexScalar;
91 typedef typename internal::conditional<FFTResultType == RealPart || FFTResultType == ImagPart, RealScalar, ComplexScalar>::type OutputScalar;
92 typedef OutputScalar CoeffReturnType;
93 typedef typename Eigen::internal::nested<TensorFFTOp>::type Nested;
94 typedef typename Eigen::internal::traits<TensorFFTOp>::StorageKind StorageKind;
95 typedef typename Eigen::internal::traits<TensorFFTOp>::Index
Index;
97 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TensorFFTOp(
const XprType& expr,
const FFT& fft)
98 : m_xpr(expr), m_fft(fft) {}
101 const FFT& fft()
const {
return m_fft; }
104 const typename internal::remove_all<typename XprType::Nested>::type& expression()
const {
109 typename XprType::Nested m_xpr;
114template <
typename FFT,
typename ArgType,
typename Device,
int FFTResultType,
int FFTDir>
115struct TensorEvaluator<const TensorFFTOp<FFT, ArgType, FFTResultType, FFTDir>, Device> {
116 typedef TensorFFTOp<FFT, ArgType, FFTResultType, FFTDir> XprType;
117 typedef typename XprType::Index Index;
118 static const int NumDims = internal::array_size<typename TensorEvaluator<ArgType, Device>::Dimensions>::value;
119 typedef DSizes<Index, NumDims> Dimensions;
120 typedef typename XprType::Scalar Scalar;
122 typedef typename std::complex<RealScalar> ComplexScalar;
123 typedef typename TensorEvaluator<ArgType, Device>::Dimensions InputDimensions;
124 typedef internal::traits<XprType> XprTraits;
125 typedef typename XprTraits::Scalar InputScalar;
126 typedef typename internal::conditional<FFTResultType == RealPart || FFTResultType == ImagPart, RealScalar, ComplexScalar>::type OutputScalar;
127 typedef OutputScalar CoeffReturnType;
128 typedef typename PacketType<OutputScalar, Device>::type PacketReturnType;
129 static const int PacketSize = internal::unpacket_traits<PacketReturnType>::size;
130 typedef StorageMemory<CoeffReturnType, Device> Storage;
131 typedef typename Storage::Type EvaluatorPointerType;
137 PreferBlockAccess =
false,
138 Layout = TensorEvaluator<ArgType, Device>::Layout,
144 typedef internal::TensorBlockNotImplemented TensorBlock;
147 EIGEN_STRONG_INLINE TensorEvaluator(
const XprType& op,
const Device& device) : m_fft(op.fft()), m_impl(op.expression(), device), m_data(NULL), m_device(device) {
148 const typename TensorEvaluator<ArgType, Device>::Dimensions& input_dims = m_impl.dimensions();
149 for (
int i = 0; i < NumDims; ++i) {
150 eigen_assert(input_dims[i] > 0);
151 m_dimensions[i] = input_dims[i];
154 if (
static_cast<int>(Layout) ==
static_cast<int>(
ColMajor)) {
156 for (
int i = 1; i < NumDims; ++i) {
157 m_strides[i] = m_strides[i - 1] * m_dimensions[i - 1];
160 m_strides[NumDims - 1] = 1;
161 for (
int i = NumDims - 2; i >= 0; --i) {
162 m_strides[i] = m_strides[i + 1] * m_dimensions[i + 1];
165 m_size = m_dimensions.TotalSize();
168 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
const Dimensions& dimensions()
const {
172 EIGEN_STRONG_INLINE
bool evalSubExprsIfNeeded(EvaluatorPointerType data) {
173 m_impl.evalSubExprsIfNeeded(NULL);
178 m_data = (EvaluatorPointerType)m_device.get((CoeffReturnType*)(m_device.allocate_temp(
sizeof(CoeffReturnType) * m_size)));
184 EIGEN_STRONG_INLINE
void cleanup() {
186 m_device.deallocate(m_data);
192 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE CoeffReturnType coeff(Index index)
const {
193 return m_data[index];
196 template <
int LoadMode>
197 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE PacketReturnType
198 packet(Index index)
const {
199 return internal::ploadt<PacketReturnType, LoadMode>(m_data + index);
202 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TensorOpCost
203 costPerCoeff(
bool vectorized)
const {
204 return TensorOpCost(
sizeof(CoeffReturnType), 0, 0, vectorized, PacketSize);
207 EIGEN_DEVICE_FUNC EvaluatorPointerType data()
const {
return m_data; }
210 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void bind(cl::sycl::handler &cgh)
const {
216 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void evalToBuf(EvaluatorPointerType data) {
217 const bool write_to_out = internal::is_same<OutputScalar, ComplexScalar>::value;
218 ComplexScalar* buf = write_to_out ? (ComplexScalar*)data : (ComplexScalar*)m_device.allocate(
sizeof(ComplexScalar) * m_size);
220 for (Index i = 0; i < m_size; ++i) {
221 buf[i] = MakeComplex<internal::is_same<InputScalar, RealScalar>::value>()(m_impl.coeff(i));
224 for (
size_t i = 0; i < m_fft.size(); ++i) {
225 Index dim = m_fft[i];
226 eigen_assert(dim >= 0 && dim < NumDims);
227 Index line_len = m_dimensions[dim];
228 eigen_assert(line_len >= 1);
229 ComplexScalar* line_buf = (ComplexScalar*)m_device.allocate(
sizeof(ComplexScalar) * line_len);
230 const bool is_power_of_two = isPowerOfTwo(line_len);
231 const Index good_composite = is_power_of_two ? 0 : findGoodComposite(line_len);
232 const Index log_len = is_power_of_two ? getLog2(line_len) : getLog2(good_composite);
234 ComplexScalar* a = is_power_of_two ? NULL : (ComplexScalar*)m_device.allocate(
sizeof(ComplexScalar) * good_composite);
235 ComplexScalar* b = is_power_of_two ? NULL : (ComplexScalar*)m_device.allocate(
sizeof(ComplexScalar) * good_composite);
236 ComplexScalar* pos_j_base_powered = is_power_of_two ? NULL : (ComplexScalar*)m_device.allocate(
sizeof(ComplexScalar) * (line_len + 1));
237 if (!is_power_of_two) {
264 for (
int j = 0; j < line_len + 1; ++j) {
265 double arg = ((EIGEN_PI * j) * j) / line_len;
266 std::complex<double> tmp(numext::cos(
arg), numext::sin(
arg));
267 pos_j_base_powered[j] =
static_cast<ComplexScalar
>(tmp);
271 for (Index partial_index = 0; partial_index < m_size / line_len; ++partial_index) {
272 const Index base_offset = getBaseOffsetFromIndex(partial_index, dim);
275 const Index stride = m_strides[dim];
277 m_device.memcpy(line_buf, &buf[base_offset], line_len*
sizeof(ComplexScalar));
279 Index offset = base_offset;
280 for (
int j = 0; j < line_len; ++j, offset += stride) {
281 line_buf[j] = buf[offset];
286 if (is_power_of_two) {
287 processDataLineCooleyTukey(line_buf, line_len, log_len);
290 processDataLineBluestein(line_buf, line_len, good_composite, log_len, a, b, pos_j_base_powered);
294 if (FFTDir == FFT_FORWARD && stride == 1) {
295 m_device.memcpy(&buf[base_offset], line_buf, line_len*
sizeof(ComplexScalar));
297 Index offset = base_offset;
298 const ComplexScalar div_factor = ComplexScalar(1.0 / line_len, 0);
299 for (
int j = 0; j < line_len; ++j, offset += stride) {
300 buf[offset] = (FFTDir == FFT_FORWARD) ? line_buf[j] : line_buf[j] * div_factor;
304 m_device.deallocate(line_buf);
305 if (!is_power_of_two) {
306 m_device.deallocate(a);
307 m_device.deallocate(b);
308 m_device.deallocate(pos_j_base_powered);
313 for (Index i = 0; i < m_size; ++i) {
314 data[i] = PartOf<FFTResultType>()(buf[i]);
316 m_device.deallocate(buf);
320 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
static bool isPowerOfTwo(Index x) {
322 return !(x & (x - 1));
326 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
static Index findGoodComposite(Index n) {
328 while (i < 2 * n - 1) i *= 2;
332 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
static Index getLog2(Index m) {
334 while (m >>= 1) log2m++;
339 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void processDataLineCooleyTukey(ComplexScalar* line_buf, Index line_len, Index log_len) {
340 eigen_assert(isPowerOfTwo(line_len));
341 scramble_FFT(line_buf, line_len);
342 compute_1D_Butterfly<FFTDir>(line_buf, line_len, log_len);
346 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void processDataLineBluestein(ComplexScalar* line_buf, Index line_len, Index good_composite, Index log_len, ComplexScalar* a, ComplexScalar* b,
const ComplexScalar* pos_j_base_powered) {
348 Index m = good_composite;
349 ComplexScalar* data = line_buf;
351 for (Index i = 0; i < n; ++i) {
352 if(FFTDir == FFT_FORWARD) {
353 a[i] = data[i] * numext::conj(pos_j_base_powered[i]);
356 a[i] = data[i] * pos_j_base_powered[i];
359 for (Index i = n; i < m; ++i) {
360 a[i] = ComplexScalar(0, 0);
363 for (Index i = 0; i < n; ++i) {
364 if(FFTDir == FFT_FORWARD) {
365 b[i] = pos_j_base_powered[i];
368 b[i] = numext::conj(pos_j_base_powered[i]);
371 for (Index i = n; i < m - n; ++i) {
372 b[i] = ComplexScalar(0, 0);
374 for (Index i = m - n; i < m; ++i) {
375 if(FFTDir == FFT_FORWARD) {
376 b[i] = pos_j_base_powered[m-i];
379 b[i] = numext::conj(pos_j_base_powered[m-i]);
384 compute_1D_Butterfly<FFT_FORWARD>(a, m, log_len);
387 compute_1D_Butterfly<FFT_FORWARD>(b, m, log_len);
389 for (Index i = 0; i < m; ++i) {
394 compute_1D_Butterfly<FFT_REVERSE>(a, m, log_len);
397 for (Index i = 0; i < m; ++i) {
401 for (Index i = 0; i < n; ++i) {
402 if(FFTDir == FFT_FORWARD) {
403 data[i] = a[i] * numext::conj(pos_j_base_powered[i]);
406 data[i] = a[i] * pos_j_base_powered[i];
411 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
static void scramble_FFT(ComplexScalar* data, Index n) {
412 eigen_assert(isPowerOfTwo(n));
414 for (Index i = 1; i < n; ++i){
416 std::swap(data[j-1], data[i-1]);
419 while (m >= 2 && j > m) {
428 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void butterfly_2(ComplexScalar* data) {
429 ComplexScalar tmp = data[1];
430 data[1] = data[0] - data[1];
435 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void butterfly_4(ComplexScalar* data) {
436 ComplexScalar tmp[4];
437 tmp[0] = data[0] + data[1];
438 tmp[1] = data[0] - data[1];
439 tmp[2] = data[2] + data[3];
440 if (Dir == FFT_FORWARD) {
441 tmp[3] = ComplexScalar(0.0, -1.0) * (data[2] - data[3]);
443 tmp[3] = ComplexScalar(0.0, 1.0) * (data[2] - data[3]);
445 data[0] = tmp[0] + tmp[2];
446 data[1] = tmp[1] + tmp[3];
447 data[2] = tmp[0] - tmp[2];
448 data[3] = tmp[1] - tmp[3];
452 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void butterfly_8(ComplexScalar* data) {
453 ComplexScalar tmp_1[8];
454 ComplexScalar tmp_2[8];
456 tmp_1[0] = data[0] + data[1];
457 tmp_1[1] = data[0] - data[1];
458 tmp_1[2] = data[2] + data[3];
459 if (Dir == FFT_FORWARD) {
460 tmp_1[3] = (data[2] - data[3]) * ComplexScalar(0, -1);
462 tmp_1[3] = (data[2] - data[3]) * ComplexScalar(0, 1);
464 tmp_1[4] = data[4] + data[5];
465 tmp_1[5] = data[4] - data[5];
466 tmp_1[6] = data[6] + data[7];
467 if (Dir == FFT_FORWARD) {
468 tmp_1[7] = (data[6] - data[7]) * ComplexScalar(0, -1);
470 tmp_1[7] = (data[6] - data[7]) * ComplexScalar(0, 1);
472 tmp_2[0] = tmp_1[0] + tmp_1[2];
473 tmp_2[1] = tmp_1[1] + tmp_1[3];
474 tmp_2[2] = tmp_1[0] - tmp_1[2];
475 tmp_2[3] = tmp_1[1] - tmp_1[3];
476 tmp_2[4] = tmp_1[4] + tmp_1[6];
478#define SQRT2DIV2 0.7071067811865476
479 if (Dir == FFT_FORWARD) {
480 tmp_2[5] = (tmp_1[5] + tmp_1[7]) * ComplexScalar(SQRT2DIV2, -SQRT2DIV2);
481 tmp_2[6] = (tmp_1[4] - tmp_1[6]) * ComplexScalar(0, -1);
482 tmp_2[7] = (tmp_1[5] - tmp_1[7]) * ComplexScalar(-SQRT2DIV2, -SQRT2DIV2);
484 tmp_2[5] = (tmp_1[5] + tmp_1[7]) * ComplexScalar(SQRT2DIV2, SQRT2DIV2);
485 tmp_2[6] = (tmp_1[4] - tmp_1[6]) * ComplexScalar(0, 1);
486 tmp_2[7] = (tmp_1[5] - tmp_1[7]) * ComplexScalar(-SQRT2DIV2, SQRT2DIV2);
488 data[0] = tmp_2[0] + tmp_2[4];
489 data[1] = tmp_2[1] + tmp_2[5];
490 data[2] = tmp_2[2] + tmp_2[6];
491 data[3] = tmp_2[3] + tmp_2[7];
492 data[4] = tmp_2[0] - tmp_2[4];
493 data[5] = tmp_2[1] - tmp_2[5];
494 data[6] = tmp_2[2] - tmp_2[6];
495 data[7] = tmp_2[3] - tmp_2[7];
499 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void butterfly_1D_merge(
500 ComplexScalar* data, Index n, Index n_power_of_2) {
504 const RealScalar wtemp = m_sin_PI_div_n_LUT[n_power_of_2];
505 const RealScalar wpi = (Dir == FFT_FORWARD)
506 ? m_minus_sin_2_PI_div_n_LUT[n_power_of_2]
507 : -m_minus_sin_2_PI_div_n_LUT[n_power_of_2];
509 const ComplexScalar wp(wtemp, wpi);
510 const ComplexScalar wp_one = wp + ComplexScalar(1, 0);
511 const ComplexScalar wp_one_2 = wp_one * wp_one;
512 const ComplexScalar wp_one_3 = wp_one_2 * wp_one;
513 const ComplexScalar wp_one_4 = wp_one_3 * wp_one;
514 const Index n2 = n / 2;
515 ComplexScalar w(1.0, 0.0);
516 for (Index i = 0; i < n2; i += 4) {
517 ComplexScalar temp0(data[i + n2] * w);
518 ComplexScalar temp1(data[i + 1 + n2] * w * wp_one);
519 ComplexScalar temp2(data[i + 2 + n2] * w * wp_one_2);
520 ComplexScalar temp3(data[i + 3 + n2] * w * wp_one_3);
523 data[i + n2] = data[i] - temp0;
526 data[i + 1 + n2] = data[i + 1] - temp1;
527 data[i + 1] += temp1;
529 data[i + 2 + n2] = data[i + 2] - temp2;
530 data[i + 2] += temp2;
532 data[i + 3 + n2] = data[i + 3] - temp3;
533 data[i + 3] += temp3;
538 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
void compute_1D_Butterfly(
539 ComplexScalar* data, Index n, Index n_power_of_2) {
540 eigen_assert(isPowerOfTwo(n));
542 compute_1D_Butterfly<Dir>(data, n / 2, n_power_of_2 - 1);
543 compute_1D_Butterfly<Dir>(data + n / 2, n / 2, n_power_of_2 - 1);
544 butterfly_1D_merge<Dir>(data, n, n_power_of_2);
546 butterfly_8<Dir>(data);
548 butterfly_4<Dir>(data);
550 butterfly_2<Dir>(data);
554 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
Index getBaseOffsetFromIndex(Index index, Index omitted_dim)
const {
557 if (
static_cast<int>(Layout) ==
static_cast<int>(
ColMajor)) {
558 for (
int i = NumDims - 1; i > omitted_dim; --i) {
559 const Index partial_m_stride = m_strides[i] / m_dimensions[omitted_dim];
560 const Index idx = index / partial_m_stride;
561 index -= idx * partial_m_stride;
562 result += idx * m_strides[i];
567 for (Index i = 0; i < omitted_dim; ++i) {
568 const Index partial_m_stride = m_strides[i] / m_dimensions[omitted_dim];
569 const Index idx = index / partial_m_stride;
570 index -= idx * partial_m_stride;
571 result += idx * m_strides[i];
579 EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
Index getIndexFromOffset(Index base, Index omitted_dim, Index offset)
const {
580 Index result = base + offset * m_strides[omitted_dim] ;
586 const FFT EIGEN_DEVICE_REF m_fft;
587 Dimensions m_dimensions;
588 array<Index, NumDims> m_strides;
589 TensorEvaluator<ArgType, Device> m_impl;
590 EvaluatorPointerType m_data;
591 const Device EIGEN_DEVICE_REF m_device;
595 const RealScalar m_sin_PI_div_n_LUT[32] = {
598 RealScalar(-0.999999999999999),
599 RealScalar(-0.292893218813453),
600 RealScalar(-0.0761204674887130),
601 RealScalar(-0.0192147195967696),
602 RealScalar(-0.00481527332780311),
603 RealScalar(-0.00120454379482761),
604 RealScalar(-3.01181303795779e-04),
605 RealScalar(-7.52981608554592e-05),
606 RealScalar(-1.88247173988574e-05),
607 RealScalar(-4.70619042382852e-06),
608 RealScalar(-1.17654829809007e-06),
609 RealScalar(-2.94137117780840e-07),
610 RealScalar(-7.35342821488550e-08),
611 RealScalar(-1.83835707061916e-08),
612 RealScalar(-4.59589268710903e-09),
613 RealScalar(-1.14897317243732e-09),
614 RealScalar(-2.87243293150586e-10),
615 RealScalar( -7.18108232902250e-11),
616 RealScalar(-1.79527058227174e-11),
617 RealScalar(-4.48817645568941e-12),
618 RealScalar(-1.12204411392298e-12),
619 RealScalar(-2.80511028480785e-13),
620 RealScalar(-7.01277571201985e-14),
621 RealScalar(-1.75319392800498e-14),
622 RealScalar(-4.38298482001247e-15),
623 RealScalar(-1.09574620500312e-15),
624 RealScalar(-2.73936551250781e-16),
625 RealScalar(-6.84841378126949e-17),
626 RealScalar(-1.71210344531737e-17),
627 RealScalar(-4.28025861329343e-18)
631 const RealScalar m_minus_sin_2_PI_div_n_LUT[32] = {
634 RealScalar(-1.00000000000000e+00),
635 RealScalar(-7.07106781186547e-01),
636 RealScalar(-3.82683432365090e-01),
637 RealScalar(-1.95090322016128e-01),
638 RealScalar(-9.80171403295606e-02),
639 RealScalar(-4.90676743274180e-02),
640 RealScalar(-2.45412285229123e-02),
641 RealScalar(-1.22715382857199e-02),
642 RealScalar(-6.13588464915448e-03),
643 RealScalar(-3.06795676296598e-03),
644 RealScalar(-1.53398018628477e-03),
645 RealScalar(-7.66990318742704e-04),
646 RealScalar(-3.83495187571396e-04),
647 RealScalar(-1.91747597310703e-04),
648 RealScalar(-9.58737990959773e-05),
649 RealScalar(-4.79368996030669e-05),
650 RealScalar(-2.39684498084182e-05),
651 RealScalar(-1.19842249050697e-05),
652 RealScalar(-5.99211245264243e-06),
653 RealScalar(-2.99605622633466e-06),
654 RealScalar(-1.49802811316901e-06),
655 RealScalar(-7.49014056584716e-07),
656 RealScalar(-3.74507028292384e-07),
657 RealScalar(-1.87253514146195e-07),
658 RealScalar(-9.36267570730981e-08),
659 RealScalar(-4.68133785365491e-08),
660 RealScalar(-2.34066892682746e-08),
661 RealScalar(-1.17033446341373e-08),
662 RealScalar(-5.85167231706864e-09),
663 RealScalar(-2.92583615853432e-09)
Namespace containing all symbols from the Eigen library.
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_arg_op< typename Derived::Scalar >, const Derived > arg(const Eigen::ArrayBase< Derived > &x)