Drudge tutorial for beginners¶
Get started¶
Drudge is a library built on top of the SymPy computer algebra library for
noncommutative and tensor alegbras. Usually for these style of problems, the
symbolic manipulation and simplification of mathematical expressions requires a
lot of context-dependent information, like the specific commutation rules and
things like the dummy symbols to be used for different ranges. So the primary
entry point for using the library is the Drudge
class, which serves
as a central repository of all kinds of domain-specific informations. To
create a drudge instance, we need to give it a Spark context so that it is
capable of parallelize things. For instance, to run things locally with all
available cores, we can do
>>> from pyspark import SparkContext
>>> spark_ctx = SparkContext('local[*]', 'drudge-tutorial')
For using Spark in cluster computing environment, please refer to the Spark documentation and setting of your cluster. With the spark context created, we can make the main entry point for drudge,
>>> import drudge
>>> dr = drudge.Drudge(spark_ctx)
Then from it, we can create the symbolic expressions as Tensor
objects, which are basically mathematical expressions containing noncommutative
objects and symbolic summations. For the noncommutativity, in spite of the
availability of some basic support of it in SymPy, here we have the
Vec
class to specifically designate the noncommutativity of its
multiplication. It can be created with a label and indexed with SymPy
expressions.
>>> v = drudge.Vec('v')
>>> import sympy
>>> a = sympy.Symbol('a')
>>> str(v[a])
'v[a]'
For the symbolic summations, we have the Range
class, which denotes
a symbolic set that a variable could be summed over. It can be created by just
a label.
>>> l = drudge.Range('L')
With these, we can create tensor objects by using the Drudge.sum()
method,
>>> x = sympy.IndexedBase('x')
>>> tensor = dr.sum((a, l), x[a] * v[a])
>>> str(tensor)
'sum_{a} x[a] * v[a]'
Now we got a symbolic tensor of a sum of vectors modulated by a SymPy IndexedBase. Actually any type of SymPy expression can be used to modulate the noncommutative vectors.
>>> tensor = dr.sum((a, l), sympy.sin(a) * v[a])
>>> str(tensor)
'sum_{a} sin(a) * v[a]'
And we can also have multiple summations and product of the vectors.
>>> b = sympy.Symbol('b')
>>> tensor = dr.sum((a, l), (b, l), x[a, b] * v[a] * v[b])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[a] * v[b]'
Of cause the multiplication of the vectors will not be commutative,
>>> tensor = dr.sum((a, l), (b, l), x[a, b] * v[b] * v[a])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[b] * v[a]'
Normally, for each symbolic range, we have some traditional symbols used as
dummies for summations over them, giving these information to drudge objects
can be very helpful. Here in this demonstration, we can use the
Drudge.set_dumms()
method.
>>> dr.set_dumms(l, sympy.symbols('a b c d'))
[a, b, c, d]
>>> dr.add_resolver_for_dumms()
where the call to the Drudge.add_resolver_for_dumms()
method could
tell the drudge to interpret all the dummy symbols to be over the range that
they are set to. By giving drudge object such domain-specific information, we
can have a lot convenience. For instance, now we can use Einstein summation
convention to create tensor object, without the need to spell all the
summations out.
>>> tensor = dr.einst(x[a, b] * v[a] * v[b])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[a] * v[b]'
Also the drudge knows what to do when more dummies are needed in mathematical operations. For instance, when we multiply things,
>>> tensor = dr.einst(x[a] * v[a])
>>> prod = tensor * tensor
>>> str(prod)
'sum_{a, b} x[a]*x[b] * v[a] * v[b]'
Here the dummy b is automatically used since the drudge object knows
available dummies for its range. Also the range and the dummies are
automatically added to the name archive of the drudge, which can be access by
Drudge.names
.
>>> p = dr.names
>>> p.L
Range('L')
>>> p.L_dumms
[a, b, c, d]
>>> p.d
d
Here in this example, we set the dummies ourselves by
Drudge.set_dumms()
. Normally, in subclasses of Drudge
for
different specific problems, such setting up is already finished within the
class. We can just directly get what we need from the names archive. There is
also a method Drudge.inject_names()
for the convenience of interactive
work.
Tensor manipulations¶
Now with tensors created by Drudge.sum()
or Drudge.einst()
, a
lot of mathematical operations are available to them. In addition to the
above example of (noncommutative) multiplication, we can also have the linear
algebraic operations of addition and scalar multiplication.
>>> tensor = dr.einst(x[a] * v[a])
>>> y = sympy.IndexedBase('y')
>>> res = tensor + dr.einst(y[a] * v[a])
>>> str(res)
'sum_{a} x[a] * v[a]\n + sum_{a} y[a] * v[a]'
>>> res = 2 * tensor
>>> str(res)
'sum_{a} 2*x[a] * v[a]'
We can also perform some complex substitutions on either the vector or the
amplitude part, by using the Drudge.subst()
method.
>>> t = sympy.IndexedBase('t')
>>> w = drudge.Vec('w')
>>> substed = tensor.subst(v[a], dr.einst(t[a, b] * w[b]))
>>> str(substed)
'sum_{a, b} x[a]*t[a, b] * w[b]'
>>> substed = tensor.subst(x[a], sympy.sin(a))
>>> str(substed)
'sum_{a} sin(a) * v[a]'
Note that here the substituted vector does not have to match the left-hand side
of the substitution exactly, pattern matching is done here. Other mathematical
operations are also available, like symbolic differentiation by
Tensor.diff()
and commutation by |
operator
Tensor.__or__()
.
Tensors are purely mathematical expressions, while the utility class
TensorDef
can be construed as tensor expressions with a left-hand
side. They can be easily created by Drudge.define()
and
Drudge.define_einst()
.
>>> v_def = dr.define_einst(v[a], t[a, b] * w[b])
>>> str(v_def)
'v[a] = sum_{b} t[a, b] * w[b]'
Their method TensorDef.act()
is like a active voice version of
Tensor.subst()
and could come handy when we need to substitute the same
definition in multiple inputs.
>>> res = v_def.act(tensor)
>>> str(res)
'sum_{a, b} x[a]*t[a, b] * w[b]'
More importantly, the definitions can be indexed directly, and the result is
designed to work well inside Drudge.sum()
or Drudge.einst()
.
For instance, for the same result, we could have,
>>> res = dr.einst(x[a] * v_def[a])
>>> str(res)
'sum_{b, a} x[a]*t[a, b] * w[b]'
When the only purpose of a vector or indexed base is to be substituted and we never intend to write tensor expressions directly in terms of them, we can just name the definition with a short name directly and put the actual base inside only. For instance,
>>> c = sympy.Symbol('c')
>>> f = dr.define_einst(sympy.IndexedBase('f')[a, b], x[a, c] * y[c, b])
>>> str(f)
'f[a, b] = sum_{c} x[a, c]*y[c, b]'
>>> str(dr.einst(f[a, a]))
'sum_{b, a} x[a, b]*y[b, a]'
which also demonstrates that the tensor definition facility can also be used for
scalar quantities. TensorDef
is also at the core of the code
optimization and generation facility in the gristmill
package.
Usually for tensorial problems, full simplification requires the utilization of
some symmetries present on the indexed quantities by permutations among their
indices. For instance, an anti-symmetric matrix entry changes sign when we
transpose the two indices. Such information can be told to drudge by using the
Drudge.set_symm()
method, by giving generators of the symmetry group
by Perm
instances. For instance, we can do,
dr.set_symm(x, drudge.Perm([1, 0], drudge.NEG))
Then the master simplification algorithm in Tensor.simplify()
is able
to take full advantage of such information.
>>> tensor = dr.einst(x[a, b] * v[a] * v[b] + x[b, a] * v[a] * v[b])
>>> str(tensor)
'sum_{a, b} x[a, b] * v[a] * v[b]\n + sum_{a, b} x[b, a] * v[a] * v[b]'
>>> str(tensor.simplify())
'0'
Normally, drudge subclasses for specific problems add symmetries for some
important indexed bases in the problem. And some drudge subclasses have helper
methods for the setting of such symmetries, like
FockDrudge.set_n_body_base()
and FockDrudge.set_dbbar_base()
.
For the simplification of the noncommutative vector parts, the base
Drudge
class does not consider any commutation rules among the
vectors. It works on the free algebra, while the subclasses could have the
specific commutation rules added for the algebraic system. For instance,
WickDrudge
add abstract commutation rules where all the commutators
have scalar values. Based on it, its special subclass FockDrudge
implements the canonical commutation relations for bosons and the canonical
anti-commutation relations for fermions. Also based on it, the subclass
CliffordDrudge
is capable of treating all kinds of Clifford
algebras, like geometric algebra, Pauli matrices, Dirac matrices, and Majorana
fermion operators. For algebraic systems where the commutator is not always a
scalar, the abstract base class GenQuadDrudge
can be used for
basically all kinds of commutation rules. For instance, its subclass
SU2LatticeDrudge
can be used for su(2) algebra in
Cartan-Weyl form.
These drudge subclasses only has the mathematical commutation rules implemented,
for convenience in solving problems, many drudge subclasses are built-in with a
lot of domain-specific information like the ranges and dummies, which are listed
in Direct support of different problems. For instance, we can easily see the commutativity of
two particle-hole excitation operators by using the PartHoleDrudge
.
>>> phdr = drudge.PartHoleDrudge(spark_ctx)
>>> t = sympy.IndexedBase('t')
>>> u = sympy.IndexedBase('u')
>>> p = phdr.names
>>> a, i = p.a, p.i
>>> excit1 = phdr.einst(t[a, i] * p.c_dag[a] * p.c_[i])
>>> excit2 = phdr.einst(u[a, i] * p.c_dag[a] * p.c_[i])
>>> comm = excit1 | excit2
>>> str(comm)
'sum_{i, a, j, b} t[a, i]*u[b, j] * c[CR, a] * c[AN, i] * c[CR, b] * c[AN, j]\n + sum_{i, a, j, b} -t[a, i]*u[b, j] * c[CR, b] * c[AN, j] * c[CR, a] * c[AN, i]'
>>> str(comm.simplify())
'0'
Note that here basically all things related to the problem, like the vector for creation and annihilation operator, the conventional dummies a and i for particle and hole labels, are directly read from the name archive of the drudge. Problem-specific drudges are supposed to give such convenience.
In addition to providing context-dependent information for general tensor operations, drudge subclasses could also provide additional operations on tensors created from them. For instance, for the above commutator, we can directly compute the expectation value with respect to the Fermi vacuum by
>>> str(comm.eval_fermi_vev())
'0'
These additional operations are called tensor methods and are documented in the drudge subclasses.
Drudge scripts¶
For maximum flexibility, drudge has been designed to be a Python library from the beginning. However, in a lot of cases, like for small tasks or for users unfamiliar with the Python language or the Spark environment, a domain-specific language capable of making simple tasks simple can be desired. Drudge script is such a language for this purpose.
A drudge script is essentially a Python script heavily tweaked to be executed
inside a special environment. So all Python lexicographical and syntactical
rules apply. For a technical description of the pre-processing and execution
drudge scripts, please see Drudge.exec_drs()
. To execute a drudge
script, we first need a Drudge
object, such that the domain specific
information about the current problem can be available. For this, we can either
have a normal Python script, where a Drudge object is created with its
Drudge.exec_drs()
called with the source code for the drudge script,
and execute it normally as Python scripts. Or drudge can also be used as the
main program, either by python3 -m drudge
or drudge
. Then two files
needs to be given as arguments. The first one is a configuration script, which
is a normal Python script with a Drudge object assigned to a special variable
DRUDGE
. Then this Drudge object will be used for the execution of the
actual drudge script given in the second argument.
As an example illustrating the basic principles and ease of drudge scripts, we
assume that we are working on a drudge with a single range registered in the
name archive as R
. To create a symbolic definition of a matrix as a product
of two matrices, suppose the drudge object can be accessed by a variable dr
,
we need to write something like:
p = dr.names
r = sympy.IndexedBase('r')
x = sympy.IndexedBase('x')
y = sympy.IndexedBase('y')
i, j, k = sympy.symbols('i j k')
def_ = dr.define(r, (i, p.R), (j, p.R), dr.sum((k, p.R), x[i, k] * y[k, j]))
which can be quite cumbersome for such a simple task. Suppose the drudge has a resolver capable of resolving any index to the range, we can write:
r = sympy.IndexedBase('r')
x = sympy.IndexedBase('x')
y = sympy.IndexedBase('y')
i, j, k = sympy.symbols('i j k')
def_ = dr.define_einst(r[i, j], x[i, k] * y[k, j])
which although is simplified a lot, still contains quite a lot of noise. Because of the Python execution model and scoping rules, the indexed bases and symbols must be explicitly created before they can be used.
Inside a drudge script, names in the name archive, all methods of the current drudge object, as well as names from the drudge, gristmill (if installed), and the SymPy package can directly be used without any qualification. More importantly, Symbol objects and IndexedBase objects are no longer needed to be explicitly created. All undefined names will be resolved as an atomic symbol, which can be construed as both a SymPy symbol and a SymPy IndexedBase. With these, the above definition can be simplified into:
def_ = define_einst(r[i, j], x[i, k] * y[k, j])
Due to the ubiquity of tensor definitions in common drudge tasks, a special
operator <<=
(Python left-shift augmented assignment operator) is introduced
for the making definitions. With this, the above definition can be written as:
r[i, j] <<= sum((k, R), x[i, k] * y[k, j])
which makes the definition and put the definition in the name archive by
Drudge.set_name()
. So by default, the definition is put into the name
archive under name r
as a TensorDef
object, and the base of the
definition is put under name _r
. Since names in the name archive do not
need to be qualified in drudge scripts:
sum((k, R), r[i, k] * r[k, j])
directly gives us the chain product XYXY. And symbolic
references to the r
tensor without the concrete definition substituted in
can still be made by using _r
, like:
s = sum((k, R), _r[i, k] * _r[k, j])
which gives us the product RR. For this, the actual definition can be substituted explicitly when desired, for example, by:
s.subst(r)
which gives us XYXY.
Note that the definition by <<=
is made by using the Drudge.def_()
method. As a result, when the drudge property Drudge.default_einst()
is set, Einstein summation convention is going to be automatically applied to
the right-hand side. So we can simply write:
r[i, j] <<= x[i, k] * y[k, j]
when the ranges of i,j,k can be resolved by the drudge.
In cases where tainting of the global name archive is undesired for a tensor
definition, we can use the <=
operator, which simply returns the definition
object without adding it to the name archive. For instance, to store the tensor
definition in a variable def_
, we can use:
def_ = r[i, j] <= x[i, k] * y[k, j]
This can be useful in functions inside drudge scripts.
Additionally, drudges could have more functions specifically to be used inside
drudge scripts. For instance, in the base Drudge
class, we have a
simple constructor S
, for converting strings to the special kind of symbols
that can be indexed and used in <<=
in drudge scripts. Also have sum_
for the actual Python built-in sum
function, which is shadowed by the
Drudge.sum()
method.
For the taste of users without much object-oriented programming, inside drudge
scripts, method calling like obj.meth(args)
can also be written as
meth(obj, args)
. For instance, for a tensor tensor
:
simplify(tensor)
is equivalent to:
tensor.simplify()
Attribute access can be done in the same way, for instance,:
n_terms(tensor)
is equivalent to:
tensor.n_terms
Note that a caveat of this syntactic sugar is that the method name cannot be defined to be anything else before the calling. For instance,:
n_terms = 10
n_terms(tensor)
does not work, since n_terms
is already defined to the integer 10, thus
cannot be called any more. Another caveat is that static methods cannot be
called in this way, which fortunately does not appear a lot in common usages of
drudge.
For the convenience of symbolic computation, all integer literals inside drudge scripts are automatically resolved to SymPy integer values, rather than the built-in integer values. As a result, we can directly write:
1 / 2
for the rational value of one-half, without having to worry about the truncation
or degradation to finite-precision floating-point numbers for Python integers.
To access built-in integers, which is normally unnecessary, we can explicitly
write something like int(1)
.
For convenience of users, some drudge functions has got slightly different
behaviour inside drudge scripts. For instance, the Tensor.simplify()
method will eagerly compute the result and repartition the terms among the
workers. And tensors also have more readable string representation inside
drudge scripts.
Examples on real-world applications¶
In this tutorial, some simple examples are run directly inside a Python
interpreter. Actually drudge is designed to work inside Jupyter notebooks as
well. By calling the Tensor.display()
method, tensor objects can be
mathematically displayed in Jupyter sessions. An example of interactive usage
of drudge, we have a sample notebook in docs/examples/ccsd.ipynb
in the
project source. Also included is a general script gencc.py
for the
automatic derivation of coupled-cluster theories, mostly to demonstrate using
drudge programmatically. And we also have a script for RCCSD theory to
demonstrate its usage in large-scale spin-explicit coupled-cluster theories.
For drudge scripts, we have two example scripts both deriving the classical CCD
theory. Both of them is based on the following configuration script
conf_ph.py
,
"""Configures a simple drudge for particle-hole model."""
from dummy_spark import SparkContext
from drudge import PartHoleDrudge
ctx = SparkContext()
dr = PartHoleDrudge(ctx)
dr.full_simplify = False
DRUDGE = dr
Here we only set a simple PartHoleDrudge
without much modification.
To illustrate the most basic usage of drudge scripts, we have example
ccd.drs
,
# A simple example on using drudge script for CCD theory
#
# In this example, the most basic aspects of using drudge scripts is
# illustrated. It should be understandable for new-comers without much
# previous Python background.
# Define the cluster excitation operator. Note that we need to inform the
# drudge that the t amplitude tensor has the double bar symmetry of
# t_{abij} = -t_{baij} = -t_{abji} = t_{baji}
set_dbbar_base(t, 2)
# Einstein summation convention can be used for easy tensor creation.
t2 = einst(
t[a, b, i, j] * c_dag[a] * c_dag[b] * c_[j] * c_[i] / 4
)
# Get the similarity-transformed Hamiltonian. Note that ``|`` operator
# computes the commutator between operators.
c0 = ham
c1 = simplify(c0 | t2)
c2 = simplify(c1 | t2)
c3 = simplify(c2 | t2)
c4 = simplify(c3 | t2)
h_bar = simplify(
c0 + c1 + (1/2) * c2 + (1/6) * c3 + (1/24) * c4
)
print('Similarity-transformed Hamiltonian has {} terms'.format(
n_terms(h_bar)
))
# Derive the working equations by projection.
en_eqn = simplify(eval_fermi_vev(h_bar))
proj = c_dag[i] * c_dag[j] * c_[b] * c_[a]
t2_eqn = simplify(eval_fermi_vev(proj * h_bar))
print('Working equation derived!')
with report('ccd.html', 'CCD theory') as rep:
rep.add('Energy equation', en_eqn)
rep.add('Doubles amplitude equation', t2_eqn)
With the comment described in the above script, we can see that drudge script
can bare a lot of resemblance to the mathematical notation. To make a
derivation of the many-body theory, we basically just use the operators like
+
, *
, and |
to do arithmetic operations on the tensors and use
simplify
to get the result simplified.
For another more advanced example, we have the ccd_adv.drs
script,
# An advanced example on using drudge script for CCD theory
#
# In this example, it is emphasized that drudge scripts are just Python scripts
# with special execution. So all Python constructions can be used for our
# convenience. At the same time, by using drudge scripts, we can have all the
# syntactical sugar for making symbolic computation easy.
#
set_dbbar_base(t, 2)
t2 = einst(
t[a, b, i, j] * c_dag[a] * c_dag[b] * c_[j] * c_[i] / 4
)
def compute_h_bar():
"""Compute the similarity transformed Hamiltonian."""
# Here we use a Python loop to get the nested commutators.
curr = ham
h_bar = ham
for order in range(0, 4):
curr = simplify(curr | t2) / (order + 1)
h_bar += curr
return simplify(h_bar)
# By using the `memoise` function, the result can be automatically dumped into
# the given pickle file, and read from it if it is already available. This can
# be convenient for large multi-step jobs.
h_bar = memoize(compute_h_bar, 'h_bar.pickle')
print('H-bar has {} terms'.format(n_terms(h_bar)))
# Derive the working equations by projection. Here we make them into tensor
# definition with explicit left-hand side, so that they can be used for
# optimization.
e <<= simplify(eval_fermi_vev(h_bar))
proj = c_dag[i] * c_dag[j] * c_[b] * c_[a]
r2[a, b, i, j] <<= simplify(eval_fermi_vev(proj * h_bar))
print('Working equation derived!')
# When the gristmill package is also installed, the evaluation of the working
# equations can also be optimized with it.
eval_seq = optimize(
[e, r2], substs={no: 1000, nv: 10000}
)
# In addition to HTML report, we can also have LaTeX report. Note that the
# report can be structured into sections with descriptions. For LaTeX output,
# the `dmath` environment from the `breqn` package can be used to break lines
# automatically inside large equations.
# Long descriptions of contents can be put in Python multi-line strings.
opt_description = """
The optimization is based on 1000 occupied orbitals and 10000 virtual orbitals,
which should be representative of common problems for CCD theory.
"""
with report('ccd.tex', 'CCD theory') as rep:
rep.add(title='Working equations')
rep.add(content=e, description='The energy equation')
rep.add(content=r2, description='Doubles amplitude equation', env='dmath')
rep.add(title='Optimized evaluation', description=opt_description)
for step in eval_seq:
rep.add(content=step, env='dmath')
In the example ccd.drs
, it is attempted to be emphasized that drudge scripts
are very similar to common mathematical notation and should be easy to get
started. In this ccd_adv.drs
example, the power and flexibility of drudge
scripts being actually Python scripts is emphasized. Foremost, rather than
spelling each order of commutation out, here the similarity-transformed
Hamiltonian ˉH is computed by using a Python loop. This
can be helpful for repetitive tasks. Also the computation of
ˉH is put inside a function. Being able to define and
execute functions makes it easy to reuse code inside drudge scripts. Here, the
function is given to the Drudge.memoize()
function. So its result is
automatically dumped into the given pickle file. When the file is already
there, the result will be directly read and used with the execution of the
function skipped. This can be helpful for large multi-step jobs.
Note that <<=
is used to make the working equations as tensor definitions of
class TensorDef
. In drudge scripts,:
variable = tensor
assigns the tensor tensor
to the variable variable
. The variable is a
normal Python variable and works in the normal Python way. And the tensor is
just a static expression of its mathematical content, with all the free symbols
being free. At the same time,:
lhs <<= tensor
defines the lhs
as the tensor, with the definition pushed into the name
archive of the drudge. By using TensorDef
objects, we also have a
left-hand side, which enables the accompanying gristmill package to optimize
the evaluation of the entire array by its advanced algorithms.
For the result, here they are written into a very structured LaTeX output, which
can be easily compiled into PDF files. Note that by using the
Report.add()
function with different arguments, we can create
structured report with sections and descriptions for the equations.
Note about importing drudge¶
In this tutorial, import drudge
and import sympy
is used and we need to
give fully-qualified name to refer to objects in them. Normally, it can be
convenient to use from drudge import *
to import everything from drudge. For
these cases, it needs to be careful that the importation of all objects from
drudge needs to follow the importation of all objects from SymPy, or the SymPy
Range
class will shallow the actual class for symbolic range in drudge.