Repository: cubed-dev/cubed
Pull request #615 (2024-11-10 11:06:04)

Change default `max_total_num_input_blocks` to 10
To allow operations to be fused as long as the total number of input blocks does not exceed this number.

Previously, the default was None, which meant operations were fused only if they had the same number of tasks.


Patch:
diff --git a/api_status.md b/api_status.md
index a71b54c7..ff1f2c59 100644
--- a/api_status.md
+++ b/api_status.md
@@ -113,8 +113,8 @@ A few of the [linear algebra extension](https://data-apis.org/array-api/2022.12/
 |                          | `qr`                | :white_check_mark: |            |                              |
 |                          | `slogdet`           | :x:                |            |                              |
 |                          | `solve`             | :x:                |            |                              |
-|                          | `svd`               | :x:                |            |                              |
-|                          | `svdvals`           | :x:                |            |                              |
+|                          | `svd`               | :white_check_mark: |            |                              |
+|                          | `svdvals`           | :white_check_mark: |            |                              |
 |                          | `tensordot`         | :white_check_mark: |            |                              |
 |                          | `trace`             | :x:                |            |                              |
 |                          | `vecdot`            | :white_check_mark: |            |                              |
diff --git a/cubed/__init__.py b/cubed/__init__.py
index 31c56240..048fea5b 100644
--- a/cubed/__init__.py
+++ b/cubed/__init__.py
@@ -15,7 +15,7 @@ config = Config(
 
 from .core.array import compute, measure_reserved_mem, visualize
 from .core.gufunc import apply_gufunc
-from .core.ops import from_array, from_zarr, map_blocks, store, to_zarr
+from .core.ops import from_array, from_zarr, map_blocks, rechunk, store, to_zarr
 from .nan_functions import nanmean, nansum
 from .overlap import map_overlap
 from .pad import pad
@@ -38,6 +38,7 @@ __all__ = [
     "nanmean",
     "nansum",
     "pad",
+    "rechunk",
     "store",
     "to_zarr",
     "visualize",
diff --git a/cubed/array_api/linalg.py b/cubed/array_api/linalg.py
index 58b6f65b..8a907a23 100644
--- a/cubed/array_api/linalg.py
+++ b/cubed/array_api/linalg.py
@@ -1,10 +1,10 @@
 from typing import NamedTuple
 
 from cubed.array_api.array_object import Array
-
-# These functions are in both the main and linalg namespaces
 from cubed.array_api.data_type_functions import result_type
 from cubed.array_api.dtypes import _floating_dtypes
+
+# These functions are in both the main and linalg namespaces
 from cubed.array_api.linear_algebra_functions import (  # noqa: F401
     matmul,
     matrix_transpose,
@@ -12,7 +12,7 @@ from cubed.array_api.linear_algebra_functions import (  # noqa: F401
     vecdot,
 )
 from cubed.backend_array_api import namespace as nxp
-from cubed.core.ops import blockwise, general_blockwise, merge_chunks
+from cubed.core.ops import blockwise, general_blockwise, merge_chunks, squeeze
 from cubed.utils import array_memory, get_item
 
 
@@ -27,6 +27,12 @@ class QRResult(NamedTuple):
     R: Array
 
 
+class SVDResult(NamedTuple):
+    U: Array
+    S: Array
+    Vh: Array
+
+
 def qr(x, /, *, mode="reduced") -> QRResult:
     if x.ndim != 2:
         raise ValueError("qr requires x to have 2 dimensions.")
@@ -43,10 +49,11 @@ def qr(x, /, *, mode="reduced") -> QRResult:
             "Consider rechunking so there is only a single column chunk."
         )
 
-    return tsqr(x)
+    Q, R, _, _, _ = tsqr(x)
+    return QRResult(Q, R)
 
 
-def tsqr(x) -> QRResult:
+def tsqr(x, compute_svd=False, finalize_svd=True):
     """Direct Tall-and-Skinny QR algorithm
 
     From:
@@ -57,18 +64,22 @@ def tsqr(x) -> QRResult:
         https://arxiv.org/abs/1301.1071
     """
 
-    # follows Algorithm 2 from Benson et al
+    # follows Algorithm 2 from Benson et al, modified for SVD
     Q1, R1 = _qr_first_step(x)
 
     if _r1_is_too_big(R1):
         R1 = _rechunk_r1(R1)
-        Q2, R2 = tsqr(R1)
+        Q2, R2, U, S, Vh = tsqr(R1, compute_svd=compute_svd, finalize_svd=False)
     else:
-        Q2, R2 = _qr_second_step(R1)
+        Q2, R2, U, S, Vh = _qr_second_step(R1, compute_svd=compute_svd)
 
     Q, R = _qr_third_step(Q1, Q2), R2
 
-    return QRResult(Q, R)
+    if compute_svd and finalize_svd:
+        U = Q @ U  # fourth step (SVD only)
+        S = squeeze(S, axis=1)  # remove extra dim
+
+    return Q, R, U, S, Vh
 
 
 def _qr_first_step(A):
@@ -108,7 +119,7 @@ def _rechunk_r1(R1, split_every=4):
     return merge_chunks(R1, chunks=chunks)
 
 
-def _qr_second_step(R1):
+def _qr_second_step(R1, compute_svd=False):
     R1_single = _merge_into_single_chunk(R1)
 
     Q2_shape = R1.shape
@@ -117,17 +128,38 @@ def _qr_second_step(R1):
     n = R1.shape[1]
     R2_shape = (n, n)
     R2_chunks = R2_shape  # single chunk
-    # qr implementation creates internal array buffers
-    extra_projected_mem = R1_single.chunkmem * 4
-    Q2, R2 = map_blocks_multiple_outputs(
-        nxp.linalg.qr,
-        R1_single,
-        shapes=[Q2_shape, R2_shape],
-        dtypes=[R1.dtype, R1.dtype],
-        chunkss=[Q2_chunks, R2_chunks],
-        extra_projected_mem=extra_projected_mem,
-    )
-    return QRResult(Q2, R2)
+
+    if not compute_svd:
+        # qr implementation creates internal array buffers
+        extra_projected_mem = R1_single.chunkmem * 4
+        Q2, R2 = map_blocks_multiple_outputs(
+            nxp.linalg.qr,
+            R1_single,
+            shapes=[Q2_shape, R2_shape],
+            dtypes=[R1.dtype, R1.dtype],
+            chunkss=[Q2_chunks, R2_chunks],
+            extra_projected_mem=extra_projected_mem,
+        )
+        return Q2, R2, None, None, None
+    else:
+        U_shape = (n, n)
+        U_chunks = U_shape
+        S_shape = (n, 1)  # extra dim since multiple outputs must have same numblocks
+        S_chunks = S_shape
+        Vh_shape = (n, n)
+        Vh_chunks = Vh_shape
+
+        # qr implementation creates internal array buffers
+        extra_projected_mem = R1_single.chunkmem * 4
+        Q2, R2, U, S, Vh = map_blocks_multiple_outputs(
+            _qr2,
+            R1_single,
+            shapes=[Q2_shape, R2_shape, U_shape, S_shape, Vh_shape],
+            dtypes=[R1.dtype, R1.dtype, R1.dtype, R1.dtype, R1.dtype],
+            chunkss=[Q2_chunks, R2_chunks, U_chunks, S_chunks, Vh_chunks],
+            extra_projected_mem=extra_projected_mem,
+        )
+        return Q2, R2, U, S, Vh
 
 
 def _merge_into_single_chunk(x, split_every=4):
@@ -138,6 +170,13 @@ def _merge_into_single_chunk(x, split_every=4):
     return x
 
 
+def _qr2(a):
+    Q, R = nxp.linalg.qr(a)
+    U, S, Vh = nxp.linalg.svd(R)
+    S = S[:, nxp.newaxis]  # add extra dim
+    return Q, R, U, S, Vh
+
+
 def _qr_third_step(Q1, Q2):
     m, n = Q1.chunksize
     k, _ = Q1.numblocks
@@ -174,6 +213,30 @@ def _q_matmul(a1, a2, q2_chunks=None, block_id=None):
     return q1 @ q2
 
 
+def svd(x, /, *, full_matrices=True) -> SVDResult:
+    if full_matrices:
+        raise ValueError("Cubed arrays only support using full_matrices=False")
+
+    nb = x.numblocks
+    # TODO: optimize case nb[0] == nb[1] == 1
+    if nb[0] > nb[1]:
+        _, _, U, S, Vh = tsqr(x, compute_svd=True)
+        truncate = x.shape[0] < x.shape[1]
+    else:
+        _, _, Vht, S, Ut = tsqr(x.T, compute_svd=True)
+        U, S, Vh = Ut.T, S, Vht.T
+        truncate = x.shape[0] > x.shape[1]
+    if truncate:  # from dask
+        k = min(x.shape)
+        U, Vh = U[:, :k], Vh[:k, :]
+    return SVDResult(U, S, Vh)
+
+
+def svdvals(x, /):
+    _, S, _ = svd(x, full_matrices=False)
+    return S
+
+
 def map_blocks_multiple_outputs(
     func,
     *args,
diff --git a/cubed/array_api/manipulation_functions.py b/cubed/array_api/manipulation_functions.py
index 284cda0e..4948d203 100644
--- a/cubed/array_api/manipulation_functions.py
+++ b/cubed/array_api/manipulation_functions.py
@@ -83,25 +83,55 @@ def concat(arrays, /, *, axis=0, chunks=None):
     if not arrays:
         raise ValueError("Need array(s) to concat")
 
+    if len({a.dtype for a in arrays}) > 1:
+        raise ValueError("concat inputs must all have the same dtype")
+
     if axis is None:
         arrays = [flatten(array) for array in arrays]
         axis = 0
 
-    # TODO: check arrays all have same shape (except in the dimension specified by axis)
-    # TODO: type promotion
-    # TODO: unify chunks
+    if len(arrays) == 1:
+        return arrays[0]
 
     a = arrays[0]
 
+    # check arrays all have same shape (except in the dimension specified by axis)
+    ndim = a.ndim
+    if not all(
+        i == axis or all(x.shape[i] == arrays[0].shape[i] for x in arrays)
+        for i in range(ndim)
+    ):
+        raise ValueError(
+            f"all the input array dimensions except for the concatenation axis must match exactly: {[x.shape for x in arrays]}"
+        )
+
+    # check arrays all have the same chunk size along axis (if more than one chunk)
+    if len({a.chunksize[axis] for a in arrays if a.numblocks[axis] > 1}) > 1:
+        raise ValueError(
+            f"all the input array chunk sizes must match along the concatenation axis: {[x.chunksize[axis] for x in arrays]}"
+        )
+
+    # unify chunks (except in the dimension specified by axis)
+    inds = [list(range(x.ndim)) for x in arrays]
+    for i, ind in enumerate(inds):
+        ind[axis] = -(i + 1)
+    uc_args = tlz.concat(zip(arrays, inds))
+    chunkss, arrays = unify_chunks(*uc_args, warn=False)
+
     # offsets along axis for the start of each array
     offsets = [0] + list(tlz.accumulate(add, [a.shape[axis] for a in arrays]))
     in_shapes = tuple(array.shape for array in arrays)
 
-    axis = validate_axis(axis, a.ndim)
+    axis = validate_axis(axis, ndim)
     shape = a.shape[:axis] + (offsets[-1],) + a.shape[axis + 1 :]
     dtype = a.dtype
     if chunks is None:
-        chunks = normalize_chunks(to_chunksize(a.chunks), shape=shape, dtype=dtype)
+        # use unified chunks except for dimension specified by axis
+        axis_chunksize = max(a.chunksize[axis] for a in arrays)
+        chunksize = tuple(
+            axis_chunksize if i == axis else chunkss[i] for i in range(ndim)
+        )
+        chunks = normalize_chunks(chunksize, shape=shape, dtype=dtype)
     else:
         chunks = normalize_chunks(chunks, shape=shape, dtype=dtype)
 
diff --git a/cubed/core/ops.py b/cubed/core/ops.py
index 23ea169b..37202559 100644
--- a/cubed/core/ops.py
+++ b/cubed/core/ops.py
@@ -775,6 +775,28 @@ def map_selection(
     max_num_input_blocks,
     **kwargs,
 ) -> "Array":
+    """
+    Apply a function to selected subsets of an input array using standard NumPy indexing notation.
+
+    Parameters
+    ----------
+    func : callable
+        Function to apply to every block to produce the output array.
+        Must accept ``block_id`` as a keyword argument (with same meaning as for ``map_blocks``).
+    selection_function : callable
+        A function that maps an output chunk key to one or more selections on the input array.
+    x: Array
+        The input array.
+    shape : tuple
+        Shape of the output array.
+    dtype : np.dtype
+        The ``dtype`` of the output array.
+    chunks : tuple
+        Chunk shape of blocks in the output array.
+    max_num_input_blocks : int
+        The maximum number of input blocks read from the input array.
+    """
+
     def key_function(out_key):
         # compute the selection on x required to get the relevant chunk for out_key
         in_sel = selection_function(out_key)
@@ -1009,6 +1031,18 @@ def map_direct(
 
 
 def rechunk(x, chunks, target_store=None):
+    """Change the chunking of an array without changing its shape or data.
+
+    Parameters
+    ----------
+    chunks : tuple
+        The desired chunks of the array after rechunking.
+
+    Returns
+    -------
+    cubed.Array
+        An array with the desired chunks.
+    """
     if isinstance(chunks, dict):
         chunks = {validate_axis(c, x.ndim): v for c, v in chunks.items()}
         for i in range(x.ndim):
@@ -1024,6 +1058,11 @@ def rechunk(x, chunks, target_store=None):
         return x
     # normalizing takes care of dict args for chunks
     target_chunks = to_chunksize(normalized_chunks)
+
+    # merge chunks special case
+    if all(c1 % c0 == 0 for c0, c1 in zip(x.chunksize, target_chunks)):
+        return merge_chunks(x, target_chunks)
+
     name = gensym()
     spec = x.spec
     if target_store is None:
diff --git a/cubed/core/optimization.py b/cubed/core/optimization.py
index 346d8e0c..5ec5064c 100644
--- a/cubed/core/optimization.py
+++ b/cubed/core/optimization.py
@@ -11,6 +11,9 @@ from cubed.primitive.blockwise import (
 
 logger = logging.getLogger(__name__)
 
+DEFAULT_MAX_TOTAL_SOURCE_ARRAYS = 4
+DEFAULT_MAX_TOTAL_NUM_INPUT_BLOCKS = 10
+
 
 def simple_optimize_dag(dag, array_names=None):
     """Apply map blocks fusion."""
@@ -154,8 +157,8 @@ def can_fuse_predecessors(
     name,
     *,
     array_names=None,
-    max_total_source_arrays=4,
-    max_total_num_input_blocks=None,
+    max_total_source_arrays=DEFAULT_MAX_TOTAL_SOURCE_ARRAYS,
+    max_total_num_input_blocks=DEFAULT_MAX_TOTAL_NUM_INPUT_BLOCKS,
     always_fuse=None,
     never_fuse=None,
 ):
@@ -242,8 +245,8 @@ def fuse_predecessors(
     name,
     *,
     array_names=None,
-    max_total_source_arrays=4,
-    max_total_num_input_blocks=None,
+    max_total_source_arrays=DEFAULT_MAX_TOTAL_SOURCE_ARRAYS,
+    max_total_num_input_blocks=DEFAULT_MAX_TOTAL_NUM_INPUT_BLOCKS,
     always_fuse=None,
     never_fuse=None,
 ):
@@ -297,8 +300,8 @@ def multiple_inputs_optimize_dag(
     dag,
     *,
     array_names=None,
-    max_total_source_arrays=4,
-    max_total_num_input_blocks=None,
+    max_total_source_arrays=DEFAULT_MAX_TOTAL_SOURCE_ARRAYS,
+    max_total_num_input_blocks=DEFAULT_MAX_TOTAL_NUM_INPUT_BLOCKS,
     always_fuse=None,
     never_fuse=None,
 ):
diff --git a/docs/api.rst b/docs/api.rst
index c2bf32b9..6afacf4f 100644
--- a/docs/api.rst
+++ b/docs/api.rst
@@ -46,6 +46,7 @@ Chunk-specific functions
     apply_gufunc
     map_blocks
     map_overlap
+    rechunk
 
 Non-standardised functions
 ==========================
diff --git a/docs/design.md b/docs/design.md
index c969ef95..e2748039 100644
--- a/docs/design.md
+++ b/docs/design.md
@@ -4,7 +4,7 @@ Cubed is composed of five layers: from the storage layer at the bottom, to the A
 
 ![Five layer diagram](images/design.svg)
 
-Blue blocks are implemented in Cubed, green in Rechunker, and red in other projects like Zarr and Beam.
+Blue blocks are implemented in Cubed; red blocks in other projects like Zarr and Lithops.
 
 Let's go through the layers from the bottom:
 
@@ -14,7 +14,7 @@ Every _array_ in Cubed is backed by a Zarr array. This means that the array type
 
 ## Runtime
 
-Cubed uses external runtimes for computation. It follows the Rechunker model (and uses its algorithm) to delegate tasks to stateless executors, which include Python (in-process), Lithops, Modal, and Apache Beam.
+Cubed uses external runtimes for computation, delegating tasks to stateless executors, which include Python (in-process), Lithops, Modal, and Apache Beam.
 
 
 ## Primitive operations
@@ -45,8 +45,7 @@ These are built on top of the primitive operations, and provide functions that a
 
     elemwise
     map_blocks
-    map_direct
-    index
+    map_selection
     reduction
     arg_reduction
 ```
diff --git a/docs/images/design.svg b/docs/images/design.svg
index 68029ea4..ad0d6976 100644
--- a/docs/images/design.svg
+++ b/docs/images/design.svg
@@ -1,1 +1,1 @@
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diff --git a/docs/images/ops.dot b/docs/images/ops.dot
index 9994a463..42532170 100644
--- a/docs/images/ops.dot
+++ b/docs/images/ops.dot
@@ -11,21 +11,20 @@ digraph {
     // core
     elemwise [style="filled"; fillcolor="#ffd8b1";];
     map_blocks [style="filled"; fillcolor="#ffd8b1";];
-    map_direct [style="filled"; fillcolor="#ffd8b1";];
+    map_selection [style="filled"; fillcolor="#ffd8b1";];
     reduction [style="filled"; fillcolor="#ffd8b1";];
     arg_reduction [style="filled"; fillcolor="#ffd8b1";];
 
     elemwise -> blockwise;
     map_blocks -> blockwise;
-    map_direct -> map_blocks;
+    map_selection -> blockwise;
     reduction -> blockwise;
-    reduction -> rechunk;
     arg_reduction -> reduction;
 
     // array API
 
     // array object
-    __getitem__ -> map_direct
+    __getitem__ -> map_selection
 
     // elementwise
     add -> elemwise
@@ -34,12 +33,11 @@ digraph {
     // linear algebra
     matmul -> blockwise;
     matmul -> reduction;
-    outer -> blockwise;
 
     // manipulation
-    concat -> map_direct;
+    concat -> blockwise;
     reshape -> rechunk;
-    reshape -> map_direct;
+    reshape -> blockwise;
     squeeze -> map_blocks;
 
     // searching
@@ -51,18 +49,17 @@ digraph {
     // utility
     all -> reduction;
 
-
     {
         rank = min;
 
         // fix horizontal placing with invisible edges
         edge[style=invis];
-        add -> negative -> outer -> matmul -> __getitem__ -> concat -> reshape -> squeeze -> argmax -> sum -> all;
+        add -> negative -> squeeze -> __getitem__ -> concat -> matmul -> sum -> all -> argmax -> reshape;
         rankdir = LR;
     }
     {
         rank = same;
-        elemwise; map_blocks; reduction;
+        elemwise; map_blocks; map_selection; reduction;
     }
     {
         rank = max;
diff --git a/docs/images/ops.dot.svg b/docs/images/ops.dot.svg
index e948db7b..0690f68b 100644
--- a/docs/images/ops.dot.svg
+++ b/docs/images/ops.dot.svg
@@ -1,244 +1,225 @@
 <?xml version="1.0" encoding="UTF-8" standalone="no"?>
 <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"
  "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
-<!-- Generated by graphviz version 2.49.3 (20211023.0002)
+<!-- Generated by graphviz version 12.1.2 (20240928.0832)
  -->
 <!-- Pages: 1 -->
-<svg width="954pt" height="260pt"
- viewBox="0.00 0.00 954.00 260.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
+<svg width="875pt" height="260pt"
+ viewBox="0.00 0.00 874.88 260.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
 <g id="graph0" class="graph" transform="scale(1 1) rotate(0) translate(4 256)">
-<polygon fill="white" stroke="transparent" points="-4,4 -4,-256 950,-256 950,4 -4,4"/>
+<polygon fill="white" stroke="none" points="-4,4 -4,-256 870.88,-256 870.88,4 -4,4"/>
 <!-- blockwise -->
 <g id="node1" class="node">
 <title>blockwise</title>
-<polygon fill="#dcbeff" stroke="black" points="331,-36 239,-36 239,0 331,0 331,-36"/>
-<text text-anchor="middle" x="285" y="-14.3" font-family="Courier,monospace" font-size="14.00">blockwise</text>
+<polygon fill="#dcbeff" stroke="black" points="477.12,-36 386.88,-36 386.88,0 477.12,0 477.12,-36"/>
+<text text-anchor="middle" x="432" y="-12.95" font-family="Courier,monospace" font-size="14.00">blockwise</text>
 </g>
 <!-- rechunk -->
 <g id="node2" class="node">
 <title>rechunk</title>
-<polygon fill="#aaffc3" stroke="black" points="731.5,-36 656.5,-36 656.5,0 731.5,0 731.5,-36"/>
-<text text-anchor="middle" x="694" y="-14.3" font-family="Courier,monospace" font-size="14.00">rechunk</text>
+<polygon fill="#aaffc3" stroke="black" points="866.88,-36 793.12,-36 793.12,0 866.88,0 866.88,-36"/>
+<text text-anchor="middle" x="830" y="-12.95" font-family="Courier,monospace" font-size="14.00">rechunk</text>
 </g>
 <!-- elemwise -->
 <g id="node3" class="node">
 <title>elemwise</title>
-<polygon fill="#ffd8b1" stroke="black" points="156,-108 72,-108 72,-72 156,-72 156,-108"/>
-<text text-anchor="middle" x="114" y="-86.3" font-family="Courier,monospace" font-size="14.00">elemwise</text>
+<polygon fill="#ffd8b1" stroke="black" points="146,-108 64,-108 64,-72 146,-72 146,-108"/>
+<text text-anchor="middle" x="105" y="-84.95" font-family="Courier,monospace" font-size="14.00">elemwise</text>
 </g>
 <!-- elemwise&#45;&gt;blockwise -->
 <g id="edge1" class="edge">
 <title>elemwise&#45;&gt;blockwise</title>
-<path fill="none" stroke="black" d="M155.83,-71.88C179.42,-62.22 209.06,-50.09 234,-39.88"/>
-<polygon fill="black" stroke="black" points="235.38,-43.1 243.3,-36.07 232.72,-36.62 235.38,-43.1"/>
+<path fill="none" stroke="black" d="M146.34,-74.45C149.25,-73.58 152.16,-72.75 155,-72 230.29,-52.1 318.94,-36.47 375.29,-27.49"/>
+<polygon fill="black" stroke="black" points="375.64,-30.97 384.97,-25.96 374.55,-24.06 375.64,-30.97"/>
 </g>
 <!-- map_blocks -->
 <g id="node4" class="node">
 <title>map_blocks</title>
-<polygon fill="#ffd8b1" stroke="black" points="547.5,-108 446.5,-108 446.5,-72 547.5,-72 547.5,-108"/>
-<text text-anchor="middle" x="497" y="-86.3" font-family="Courier,monospace" font-size="14.00">map_blocks</text>
+<polygon fill="#ffd8b1" stroke="black" points="262.25,-108 163.75,-108 163.75,-72 262.25,-72 262.25,-108"/>
+<text text-anchor="middle" x="213" y="-84.95" font-family="Courier,monospace" font-size="14.00">map_blocks</text>
 </g>
 <!-- map_blocks&#45;&gt;blockwise -->
 <g id="edge2" class="edge">
 <title>map_blocks&#45;&gt;blockwise</title>
-<path fill="none" stroke="black" d="M446.49,-72.32C414.75,-61.84 373.76,-48.31 341.02,-37.5"/>
-<polygon fill="black" stroke="black" points="341.74,-34.05 331.14,-34.24 339.54,-40.69 341.74,-34.05"/>
+<path fill="none" stroke="black" d="M262.69,-73.12C296.43,-62.33 341.1,-48.06 376.08,-36.87"/>
+<polygon fill="black" stroke="black" points="376.7,-40.35 385.16,-33.97 374.57,-33.68 376.7,-40.35"/>
 </g>
-<!-- map_direct -->
+<!-- map_selection -->
 <g id="node5" class="node">
-<title>map_direct</title>
-<polygon fill="#ffd8b1" stroke="black" points="547.5,-180 446.5,-180 446.5,-144 547.5,-144 547.5,-180"/>
-<text text-anchor="middle" x="497" y="-158.3" font-family="Courier,monospace" font-size="14.00">map_direct</text>
+<title>map_selection</title>
+<polygon fill="#ffd8b1" stroke="black" points="403.62,-108 280.38,-108 280.38,-72 403.62,-72 403.62,-108"/>
+<text text-anchor="middle" x="342" y="-84.95" font-family="Courier,monospace" font-size="14.00">map_selection</text>
 </g>
-<!-- map_direct&#45;&gt;map_blocks -->
+<!-- map_selection&#45;&gt;blockwise -->
 <g id="edge3" class="edge">
-<title>map_direct&#45;&gt;map_blocks</title>
-<path fill="none" stroke="black" d="M497,-143.7C497,-135.98 497,-126.71 497,-118.11"/>
-<polygon fill="black" stroke="black" points="500.5,-118.1 497,-108.1 493.5,-118.1 500.5,-118.1"/>
+<title>map_selection&#45;&gt;blockwise</title>
+<path fill="none" stroke="black" d="M364.25,-71.7C375.28,-63.11 388.79,-52.61 400.84,-43.24"/>
+<polygon fill="black" stroke="black" points="402.76,-46.18 408.5,-37.28 398.46,-40.65 402.76,-46.18"/>
 </g>
 <!-- reduction -->
 <g id="node6" class="node">
 <title>reduction</title>
-<polygon fill="#ffd8b1" stroke="black" points="783,-108 691,-108 691,-72 783,-72 783,-108"/>
-<text text-anchor="middle" x="737" y="-86.3" font-family="Courier,monospace" font-size="14.00">reduction</text>
+<polygon fill="#ffd8b1" stroke="black" points="619.12,-108 528.88,-108 528.88,-72 619.12,-72 619.12,-108"/>
+<text text-anchor="middle" x="574" y="-84.95" font-family="Courier,monospace" font-size="14.00">reduction</text>
 </g>
 <!-- reduction&#45;&gt;blockwise -->
 <g id="edge4" class="edge">
 <title>reduction&#45;&gt;blockwise</title>
-<path fill="none" stroke="black" d="M690.75,-81.84C607.23,-68.9 431.55,-41.7 341.23,-27.71"/>
-<polygon fill="black" stroke="black" points="341.6,-24.22 331.18,-26.15 340.53,-31.14 341.6,-24.22"/>
-</g>
-<!-- reduction&#45;&gt;rechunk -->
-<g id="edge5" class="edge">
-<title>reduction&#45;&gt;rechunk</title>
-<path fill="none" stroke="black" d="M726.37,-71.7C721.37,-63.56 715.31,-53.69 709.79,-44.7"/>
-<polygon fill="black" stroke="black" points="712.72,-42.79 704.51,-36.1 706.76,-46.46 712.72,-42.79"/>
+<path fill="none" stroke="black" d="M538.53,-71.52C519.93,-62.34 496.93,-51.01 477.07,-41.22"/>
+<polygon fill="black" stroke="black" points="478.89,-38.21 468.37,-36.93 475.79,-44.49 478.89,-38.21"/>
 </g>
 <!-- arg_reduction -->
 <g id="node7" class="node">
 <title>arg_reduction</title>
-<polygon fill="#ffd8b1" stroke="black" points="812,-180 686,-180 686,-144 812,-144 812,-180"/>
-<text text-anchor="middle" x="749" y="-158.3" font-family="Courier,monospace" font-size="14.00">arg_reduction</text>
+<polygon fill="#ffd8b1" stroke="black" points="763.62,-180 640.38,-180 640.38,-144 763.62,-144 763.62,-180"/>
+<text text-anchor="middle" x="702" y="-156.95" font-family="Courier,monospace" font-size="14.00">arg_reduction</text>
 </g>
 <!-- arg_reduction&#45;&gt;reduction -->
-<g id="edge6" class="edge">
+<g id="edge5" class="edge">
 <title>arg_reduction&#45;&gt;reduction</title>
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+<title>__getitem__&#45;&gt;map_selection</title>
+<path fill="none" stroke="black" d="M320.07,-215.59C324.29,-191.61 331.94,-148.14 337,-119.42"/>
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 <title>add</title>
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 </g>
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+<g id="edge7" class="edge">
 <title>add&#45;&gt;elemwise</title>
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+<g id="edge8" class="edge">
 <title>negative&#45;&gt;elemwise</title>
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-<polygon fill="black" stroke="black" points="117.5,-118.19 114,-108.19 110.5,-118.19 117.5,-118.19"/>
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+<polygon fill="black" stroke="black" points="110.11,-119.41 106.05,-109.62 103.12,-119.8 110.11,-119.41"/>
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-<g id="node12" class="node">
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-<text text-anchor="middle" x="204" y="-230.3" font-family="Courier,monospace" font-size="14.00">outer</text>
+<!-- squeeze -->
+<g id="node14" class="node">
+<title>squeeze</title>
+<polygon fill="none" stroke="black" points="245.88,-252 172.12,-252 172.12,-216 245.88,-216 245.88,-252"/>
+<text text-anchor="middle" x="209" y="-228.95" font-family="Courier,monospace" font-size="14.00">squeeze</text>
 </g>
-<!-- negative&#45;&gt;outer -->
+<!-- negative&#45;&gt;squeeze -->
 <!-- matmul -->
 <g id="node11" class="node">
 <title>matmul</title>
-<polygon fill="none" stroke="black" points="318.5,-252 251.5,-252 251.5,-216 318.5,-216 318.5,-252"/>
-<text text-anchor="middle" x="285" y="-230.3" font-family="Courier,monospace" font-size="14.00">matmul</text>
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+<text text-anchor="middle" x="505" y="-228.95" font-family="Courier,monospace" font-size="14.00">matmul</text>
 </g>
 <!-- matmul&#45;&gt;blockwise -->
-<g id="edge10" class="edge">
+<g id="edge9" class="edge">
 <title>matmul&#45;&gt;blockwise</title>
-<path fill="none" stroke="black" d="M285,-215.85C285,-178.83 285,-91.18 285,-46.39"/>
-<polygon fill="black" stroke="black" points="288.5,-46.23 285,-36.23 281.5,-46.23 288.5,-46.23"/>
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+<polygon fill="black" stroke="black" points="444.91,-46 438.37,-37.66 438.28,-48.26 444.91,-46"/>
 </g>
 <!-- matmul&#45;&gt;reduction -->
-<g id="edge11" class="edge">
+<g id="edge10" class="edge">
 <title>matmul&#45;&gt;reduction</title>
-<path fill="none" stroke="black" d="M308.16,-215.87C336.89,-195.5 388.06,-161.99 437,-144 518.65,-113.98 619.23,-100.66 680.7,-94.97"/>
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+<polygon fill="black" stroke="black" points="563.8,-120 565.02,-109.47 557.51,-116.94 563.8,-120"/>
 </g>
-<!-- matmul&#45;&gt;__getitem__ -->
-<!-- outer&#45;&gt;blockwise -->
-<g id="edge12" class="edge">
-<title>outer&#45;&gt;blockwise</title>
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-<polygon fill="black" stroke="black" points="278.21,-46.82 278.48,-36.23 271.66,-44.35 278.21,-46.82"/>
+<!-- sum -->
+<g id="node16" class="node">
+<title>sum</title>
+<polygon fill="none" stroke="black" points="610,-252 556,-252 556,-216 610,-216 610,-252"/>
+<text text-anchor="middle" x="583" y="-228.95" font-family="Courier,monospace" font-size="14.00">sum</text>
 </g>
-<!-- outer&#45;&gt;matmul -->
-<!-- concat&#45;&gt;map_direct -->
-<g id="edge13" class="edge">
-<title>concat&#45;&gt;map_direct</title>
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-<polygon fill="black" stroke="black" points="500.5,-190.1 497,-180.1 493.5,-190.1 500.5,-190.1"/>
+<!-- matmul&#45;&gt;sum -->
+<!-- concat&#45;&gt;blockwise -->
+<g id="edge11" class="edge">
+<title>concat&#45;&gt;blockwise</title>
+<path fill="none" stroke="black" d="M421.88,-215.85C423.77,-179.14 428.21,-92.66 430.53,-47.53"/>
+<polygon fill="black" stroke="black" points="434.02,-47.91 431.04,-37.74 427.03,-47.55 434.02,-47.91"/>
 </g>
+<!-- concat&#45;&gt;matmul -->
 <!-- reshape -->
-<g id="node14" class="node">
+<g id="node13" class="node">
 <title>reshape</title>
-<polygon fill="none" stroke="black" points="623.5,-252 548.5,-252 548.5,-216 623.5,-216 623.5,-252"/>
-<text text-anchor="middle" x="586" y="-230.3" font-family="Courier,monospace" font-size="14.00">reshape</text>
+<polygon fill="none" stroke="black" points="857.88,-252 784.12,-252 784.12,-216 857.88,-216 857.88,-252"/>
+<text text-anchor="middle" x="821" y="-228.95" font-family="Courier,monospace" font-size="14.00">reshape</text>
+</g>
+<!-- reshape&#45;&gt;blockwise -->
+<g id="edge13" class="edge">
+<title>reshape&#45;&gt;blockwise</title>
+<path fill="none" stroke="black" d="M815.27,-215.6C808.08,-196.04 794.08,-164.38 773,-144 693.45,-67.09 562.72,-36.88 488.36,-25.49"/>
+<polygon fill="black" stroke="black" points="489.27,-22.09 478.87,-24.1 488.25,-29.01 489.27,-22.09"/>
 </g>
-<!-- concat&#45;&gt;reshape -->
 <!-- reshape&#45;&gt;rechunk -->
-<g id="edge14" class="edge">
+<g id="edge12" class="edge">
 <title>reshape&#45;&gt;rechunk</title>
-<path fill="none" stroke="black" d="M594.66,-215.85C613.5,-178.52 658.32,-89.69 680.75,-45.26"/>
-<polygon fill="black" stroke="black" points="683.92,-46.74 685.3,-36.23 677.67,-43.58 683.92,-46.74"/>
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+<polygon fill="black" stroke="black" points="832.29,-47.88 829.21,-37.74 825.29,-47.59 832.29,-47.88"/>
 </g>
-<!-- reshape&#45;&gt;map_direct -->
-<g id="edge15" class="edge">
-<title>reshape&#45;&gt;map_direct</title>
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-<polygon fill="black" stroke="black" points="528.77,-183.54 518.75,-180.1 524.44,-189.04 528.77,-183.54"/>
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-<!-- squeeze -->
-<g id="node15" class="node">
-<title>squeeze</title>
-<polygon fill="none" stroke="black" points="716.5,-252 641.5,-252 641.5,-216 716.5,-216 716.5,-252"/>
-<text text-anchor="middle" x="679" y="-230.3" font-family="Courier,monospace" font-size="14.00">squeeze</text>
-</g>
-<!-- reshape&#45;&gt;squeeze -->
 <!-- squeeze&#45;&gt;map_blocks -->
-<g id="edge16" class="edge">
+<g id="edge14" class="edge">
 <title>squeeze&#45;&gt;map_blocks</title>
-<path fill="none" stroke="black" d="M657.05,-215.87C624.66,-190.6 563.97,-143.25 527.35,-114.68"/>
-<polygon fill="black" stroke="black" points="529.07,-111.58 519.03,-108.19 524.76,-117.1 529.07,-111.58"/>
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+<polygon fill="black" stroke="black" points="215.69,-119.72 212.48,-109.62 208.7,-119.52 215.69,-119.72"/>
 </g>
+<!-- squeeze&#45;&gt;__getitem__ -->
 <!-- argmax -->
-<g id="node16" class="node">
+<g id="node15" class="node">
 <title>argmax</title>
-<polygon fill="none" stroke="black" points="801.5,-252 734.5,-252 734.5,-216 801.5,-216 801.5,-252"/>
-<text text-anchor="middle" x="768" y="-230.3" font-family="Courier,monospace" font-size="14.00">argmax</text>
+<polygon fill="none" stroke="black" points="765.75,-252 700.25,-252 700.25,-216 765.75,-216 765.75,-252"/>
+<text text-anchor="middle" x="733" y="-228.95" font-family="Courier,monospace" font-size="14.00">argmax</text>
 </g>
-<!-- squeeze&#45;&gt;argmax -->
 <!-- argmax&#45;&gt;arg_reduction -->
-<g id="edge17" class="edge">
+<g id="edge15" class="edge">
 <title>argmax&#45;&gt;arg_reduction</title>
-<path fill="none" stroke="black" d="M763.3,-215.7C761.19,-207.9 758.64,-198.51 756.28,-189.83"/>
-<polygon fill="black" stroke="black" points="759.64,-188.84 753.64,-180.1 752.88,-190.67 759.64,-188.84"/>
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-<title>sum</title>
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-<text text-anchor="middle" x="847" y="-230.3" font-family="Courier,monospace" font-size="14.00">sum</text>
+<path fill="none" stroke="black" d="M725.34,-215.7C721.96,-208.07 717.91,-198.92 714.13,-190.4"/>
+<polygon fill="black" stroke="black" points="717.44,-189.21 710.19,-181.49 711.04,-192.05 717.44,-189.21"/>
 </g>
-<!-- argmax&#45;&gt;sum -->
+<!-- argmax&#45;&gt;reshape -->
 <!-- sum&#45;&gt;reduction -->
-<g id="edge18" class="edge">
+<g id="edge16" class="edge">
 <title>sum&#45;&gt;reduction</title>
-<path fill="none" stroke="black" d="M845.63,-215.99C843.32,-196.8 837.18,-165.46 821,-144 811.55,-131.46 798.19,-121.14 784.91,-113.04"/>
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+<polygon fill="black" stroke="black" points="579.31,-119.38 575.18,-109.62 572.32,-119.82 579.31,-119.38"/>
 </g>
 <!-- all -->
-<g id="node18" class="node">
+<g id="node17" class="node">
 <title>all</title>
-<polygon fill="none" stroke="black" points="946,-252 892,-252 892,-216 946,-216 946,-252"/>
-<text text-anchor="middle" x="919" y="-230.3" font-family="Courier,monospace" font-size="14.00">all</text>
+<polygon fill="none" stroke="black" points="682,-252 628,-252 628,-216 682,-216 682,-252"/>
+<text text-anchor="middle" x="655" y="-228.95" font-family="Courier,monospace" font-size="14.00">all</text>
 </g>
 <!-- sum&#45;&gt;all -->
 <!-- all&#45;&gt;reduction -->
-<g id="edge19" class="edge">
+<g id="edge17" class="edge">
 <title>all&#45;&gt;reduction</title>
-<path fill="none" stroke="black" d="M907.52,-215.79C893.92,-196.41 869.73,-164.91 843,-144 826.99,-131.48 807.56,-120.73 789.72,-112.25"/>
-<polygon fill="black" stroke="black" points="791.11,-109.03 780.57,-108.02 788.18,-115.39 791.11,-109.03"/>
+<path fill="none" stroke="black" d="M645.07,-215.59C631.15,-191.18 605.71,-146.58 589.34,-117.89"/>
+<polygon fill="black" stroke="black" points="592.51,-116.38 584.51,-109.43 586.43,-119.85 592.51,-116.38"/>
 </g>
+<!-- all&#45;&gt;argmax -->
 </g>
 </svg>
diff --git a/docs/operations.md b/docs/operations.md
index e7b003d8..7865fdb0 100644
--- a/docs/operations.md
+++ b/docs/operations.md
@@ -39,18 +39,20 @@ This example shows how the `squeeze` operation is implemented using `map_blocks`
 
 ![The map_blocks core operation](images/map_blocks.svg)
 
-## `map_direct`
+## `map_selection`
 
-The `map_direct` operation is more general than `map_blocks` since it allows input arrays to be read directly, without regard for block boundaries.
+The `map_selection` operation selects subsets of an input array using standard NumPy indexing notation.
 
 * No input array attributes are preserved in general
-* __Multiple__ inputs, __single__ output
+* __Single__ input, __single__ output
 
-It is a core operation that is implemented using `map_blocks`, which is in turn implemented using `blockwise`. It works by creating an empty array that has the same shape and chunk structure as the output, and calling `map_blocks` on this empty array, passing in the input arrays as side inputs to the function, which may access them in whatever way is needed.
+It is a core operation that is implemented using `blockwise` on the output's blocks. It works by converting indexing selections, such as slices, to keys that refer to blocks in the input array, then retrieving these blocks, slicing them, and assembling them into the final output block.
 
-This example shows how `concat` is implemented using `map_direct`. Each block in the output array is read directly from one or more blocks from the inputs.
+This example shows how `index` is implemented using `map_selection`. Each block in the output array is read directly from one or more blocks from the inputs.
 
-![The map_direct core operation](images/map_direct.svg)
+![The map_selection core operation](images/map_selection.svg)
+
+Note: previously, operations that now use `map_selection` were written using `map_direct`, which allowed input arrays to be read directly. The main difference between the two operations is that `map_selection` tracks which blocks are read by the operation, which allows the optimizer to fuse operations. This is unlike `map_direct`, which does not provide information about block inputs, and therefore cannot be fused by the optimizer.
 
 ## `blockwise`
 
@@ -63,6 +65,13 @@ This example shows how `outer` is implemented using `blockwise`. Each block from
 
 ![The blockwise primitive operation](images/blockwise.svg)
 
+Note: the `general_blockwise` operation is a more general form of `blockwise` that uses a function to specify the block mapping, rather than an index notation, and which supports multiple outputs.
+
+* No input array attributes are preserved in general
+* __Multiple__ inputs, __multiple__ outputs
+
+For multiple outputs, all output arrays must have matching `numblocks`.
+
 (rechunk-operation)=
 ## `rechunk`
 
diff --git a/docs/user-guide/optimization.md b/docs/user-guide/optimization.md
index 3844d581..ec556546 100644
--- a/docs/user-guide/optimization.md
+++ b/docs/user-guide/optimization.md
@@ -81,9 +81,9 @@ e.visualize()
 The output explains which operations can or can't be fused, and why:
 
 ```
-DEBUG:cubed.core.optimization:can't fuse op-001 since it is not a primitive operation, or it uses map_direct
-DEBUG:cubed.core.optimization:can't fuse op-002 since it is not a primitive operation, or it uses map_direct
-DEBUG:cubed.core.optimization:can't fuse op-003 since it is not a primitive operation, or it uses map_direct
+DEBUG:cubed.core.optimization:can't fuse op-001 since it is not a primitive operation, or it uses an operation that can't be fused (concat or stack)
+DEBUG:cubed.core.optimization:can't fuse op-002 since it is not a primitive operation, or it uses an operation that can't be fused (concat or stack)
+DEBUG:cubed.core.optimization:can't fuse op-003 since it is not a primitive operation, or it uses an operation that can't be fused (concat or stack)
 DEBUG:cubed.core.optimization:can't fuse op-004 since no predecessor ops can be fused
 DEBUG:cubed.primitive.blockwise:can fuse op-005 since num tasks of predecessor ops match
 ```
@@ -112,9 +112,9 @@ e.visualize(optimize_function=opt_fn)
 
 The `max_total_num_input_blocks` argument to `multiple_inputs_optimize_dag` specifies the maximum number of input blocks (chunks) that are allowed in the fused operation.
 
-Again, this is to limit the number of reads that an individual task must perform. The default is `None`, which means that operations are fused only if they have the same number of tasks. If set to an integer, then this limitation is removed, and tasks with a different number of tasks will be fused - as long as the total number of input blocks does not exceed the maximum. This setting is useful for reductions, and can be set using `functools.partial`:
+Again, this is to limit the number of reads that an individual task must perform. If set to `None`, operations are fused only if they have the same number of tasks. If set to an integer (the default is 10), then tasks with a different number of tasks will be fused - as long as the total number of input blocks does not exceed the maximum. This setting is useful for reductions, and can be changed using `functools.partial`:
 
 ```python
-opt_fn = partial(multiple_inputs_optimize_dag, max_total_num_input_blocks=10)
+opt_fn = partial(multiple_inputs_optimize_dag, max_total_num_input_blocks=20)
 e.visualize(optimize_function=opt_fn)
 ```


Test patch:
diff --git a/cubed/tests/test_array_api.py b/cubed/tests/test_array_api.py
index 42a245cb..4ea6653c 100644
--- a/cubed/tests/test_array_api.py
+++ b/cubed/tests/test_array_api.py
@@ -506,6 +506,43 @@ def test_concat(spec, executor):
     )
 
 
+def test_concat_different_chunks(spec):
+    a = xp.asarray([[1], [5]], chunks=(2, 2), spec=spec)
+    b = xp.asarray([[2, 3, 4], [6, 7, 8]], chunks=(2, 3), spec=spec)
+    c = xp.concat([a, b], axis=1)
+    assert_array_equal(
+        c.compute(),
+        np.concatenate(
+            [
+                np.array([[1], [5]]),
+                np.array([[2, 3, 4], [6, 7, 8]]),
+            ],
+            axis=1,
+        ),
+    )
+
+
+@pytest.mark.parametrize("axis", [None, 0])
+def test_concat_single_array(spec, axis):
+    a = xp.full((4, 5), 1, chunks=(3, 2), spec=spec)
+    d = xp.concat([a], axis=axis)
+    assert_array_equal(
+        d.compute(),
+        np.concatenate([np.full((4, 5), 1)], axis=axis),
+    )
+
+
+def test_concat_incompatible_shapes(spec):
+    a = xp.full((4, 5), 1, chunks=(3, 2), spec=spec)
+    b = xp.full((4, 6), 2, chunks=(3, 2), spec=spec)
+    with pytest.raises(
+        ValueError,
+        match="all the input array dimensions except for the concatenation axis must match exactly",
+    ):
+        xp.concat([a, b], axis=0)
+    xp.concat([a, b], axis=1)  # OK
+
+
 def test_expand_dims(spec, executor):
     a = xp.asarray([1, 2, 3], chunks=(2,), spec=spec)
     b = xp.expand_dims(a, axis=0)
diff --git a/cubed/tests/test_core.py b/cubed/tests/test_core.py
index ba916517..efba6ffe 100644
--- a/cubed/tests/test_core.py
+++ b/cubed/tests/test_core.py
@@ -298,6 +298,7 @@ def test_rechunk(spec, executor, new_chunks, expected_chunks):
 def test_rechunk_same_chunks(spec):
     a = xp.asarray([[1, 2, 3], [4, 5, 6], [7, 8, 9]], chunks=(2, 1), spec=spec)
     b = a.rechunk((2, 1))
+    assert b is a
     task_counter = TaskCounter()
     res = b.compute(callbacks=[task_counter])
     # no tasks should have run since chunks are same
@@ -314,6 +315,31 @@ def test_rechunk_intermediate(tmp_path):
     assert_array_equal(b.compute(), np.ones((4, 4)))
     intermediates = [n for (n, d) in b.plan.dag.nodes(data=True) if "-int" in d["name"]]
     assert len(intermediates) == 1
+    rechunks = [
+        n
+        for (n, d) in b.plan.dag.nodes(data=True)
+        if d.get("op_name", None) == "rechunk"
+    ]
+    assert len(rechunks) > 0
+
+
+def test_rechunk_merge_chunks_optimization():
+    a = xp.asarray(
+        [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]],
+        chunks=(2, 1),
+    )
+    b = a.rechunk((4, 2))
+    assert b.chunks == ((4,), (2, 2))
+    assert_array_equal(
+        b.compute(),
+        np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]),
+    )
+    rechunks = [
+        n
+        for (n, d) in b.plan.dag.nodes(data=True)
+        if d.get("op_name", None) == "rechunk"
+    ]
+    assert len(rechunks) == 0
 
 
 def test_compute_is_idempotent(spec, executor):
diff --git a/cubed/tests/test_linalg.py b/cubed/tests/test_linalg.py
index 0ef72636..585a72d4 100644
--- a/cubed/tests/test_linalg.py
+++ b/cubed/tests/test_linalg.py
@@ -57,3 +57,47 @@ def test_qr_chunking():
         match=r"qr only supports tall-and-skinny \(single column chunk\) arrays.",
     ):
         xp.linalg.qr(A)
+
+
+def test_svd():
+    A = np.reshape(np.arange(32, dtype=np.float64), (16, 2))
+
+    U, S, Vh = xp.linalg.svd(xp.asarray(A, chunks=(4, 2)), full_matrices=False)
+    U, S, Vh = cubed.compute(U, S, Vh)
+
+    assert_allclose(U * S @ Vh, A, atol=1e-08)
+    assert_allclose(U.T @ U, np.eye(2, 2), atol=1e-08)  # U must be orthonormal
+    assert_allclose(Vh @ Vh.T, np.eye(2, 2), atol=1e-08)  # Vh must be orthonormal
+
+
+def test_svd_recursion():
+    A = np.reshape(np.arange(128, dtype=np.float64), (64, 2))
+
+    # find a memory setting where recursion happens
+    found = False
+    for factor in range(4, 16):
+        spec = cubed.Spec(allowed_mem=128 * factor, reserved_mem=0)
+
+        try:
+            U, S, Vh = xp.linalg.svd(
+                xp.asarray(A, chunks=(8, 2), spec=spec), full_matrices=False
+            )
+
+            found = True
+            plan_unopt = arrays_to_plan(U, S, Vh)._finalize()
+            assert plan_unopt.num_primitive_ops() > 4  # more than without recursion
+
+            U, S, Vh = cubed.compute(U, S, Vh)
+
+            assert_allclose(U * S @ Vh, A, atol=1e-08)
+            assert_allclose(U.T @ U, np.eye(2, 2), atol=1e-08)  # U must be orthonormal
+            assert_allclose(
+                Vh @ Vh.T, np.eye(2, 2), atol=1e-08
+            )  # Vh must be orthonormal
+
+            break
+
+        except ValueError:
+            pass  # not enough memory
+
+    assert found
diff --git a/cubed/tests/test_optimization.py b/cubed/tests/test_optimization.py
index 6d635f0c..ebb94022 100644
--- a/cubed/tests/test_optimization.py
+++ b/cubed/tests/test_optimization.py
@@ -11,6 +11,8 @@ import cubed.random
 from cubed.backend_array_api import namespace as nxp
 from cubed.core.ops import elemwise, merge_chunks, partial_reduce
 from cubed.core.optimization import (
+    DEFAULT_MAX_TOTAL_NUM_INPUT_BLOCKS,
+    DEFAULT_MAX_TOTAL_SOURCE_ARRAYS,
     fuse_all_optimize_dag,
     fuse_only_optimize_dag,
     fuse_predecessors,
@@ -223,7 +225,11 @@ def fuse_one_level(arr, *, always_fuse=None):
     )
 
 
-def fuse_multiple_levels(*, max_total_source_arrays=4, max_total_num_input_blocks=None):
+def fuse_multiple_levels(
+    *,
+    max_total_source_arrays=DEFAULT_MAX_TOTAL_SOURCE_ARRAYS,
+    max_total_num_input_blocks=DEFAULT_MAX_TOTAL_NUM_INPUT_BLOCKS,
+):
     # use multiple_inputs_optimize_dag to test multiple levels of fusion
     return partial(
         multiple_inputs_optimize_dag,
@@ -899,8 +905,7 @@ def test_fuse_merge_chunks_unary(spec):
     b = xp.negative(a)
     c = merge_chunks(b, chunks=(3, 2))
 
-    # specify max_total_num_input_blocks to force c to fuse
-    opt_fn = fuse_multiple_levels(max_total_num_input_blocks=3)
+    opt_fn = fuse_multiple_levels()
 
     c.visualize(optimize_function=opt_fn)
 
@@ -921,6 +926,16 @@ def test_fuse_merge_chunks_unary(spec):
     result = c.compute(optimize_function=opt_fn)
     assert_array_equal(result, -np.ones((3, 2)))
 
+    # now set max_total_num_input_blocks=None which means
+    # "only fuse if ops have same number of tasks", which they don't here
+    opt_fn = fuse_multiple_levels(max_total_num_input_blocks=None)
+    optimized_dag = c.plan.optimize(optimize_function=opt_fn).dag
+
+    # merge_chunks uses a hidden op and array for block ids - ignore when comparing structure
+    assert not structurally_equivalent(
+        optimized_dag, expected_fused_dag, remove_hidden=True
+    )
+
 
 # merge chunks with different number of tasks (c has more tasks than d)
 #
@@ -936,8 +951,7 @@ def test_fuse_merge_chunks_binary(spec):
     c = xp.add(a, b)
     d = merge_chunks(c, chunks=(3, 2))
 
-    # specify max_total_num_input_blocks to force d to fuse
-    opt_fn = fuse_multiple_levels(max_total_num_input_blocks=6)
+    opt_fn = fuse_multiple_levels()
 
     d.visualize(optimize_function=opt_fn)
 
@@ -963,6 +977,16 @@ def test_fuse_merge_chunks_binary(spec):
     result = d.compute(optimize_function=opt_fn)
     assert_array_equal(result, 2 * np.ones((3, 2)))
 
+    # now set max_total_num_input_blocks=None which means
+    # "only fuse if ops have same number of tasks", which they don't here
+    opt_fn = fuse_multiple_levels(max_total_num_input_blocks=None)
+    optimized_dag = d.plan.optimize(optimize_function=opt_fn).dag
+
+    # merge_chunks uses a hidden op and array for block ids - ignore when comparing structure
+    assert not structurally_equivalent(
+        optimized_dag, expected_fused_dag, remove_hidden=True
+    )
+
 
 # like test_fuse_merge_chunks_unary, except uses partial_reduce
 def test_fuse_partial_reduce_unary(spec):
@@ -970,8 +994,7 @@ def test_fuse_partial_reduce_unary(spec):
     b = xp.negative(a)
     c = partial_reduce(b, np.sum, split_every={0: 3})
 
-    # specify max_total_num_input_blocks to force c to fuse
-    opt_fn = fuse_multiple_levels(max_total_num_input_blocks=3)
+    opt_fn = fuse_multiple_levels()
 
     c.visualize(optimize_function=opt_fn)
 
@@ -996,8 +1019,7 @@ def test_fuse_partial_reduce_binary(spec):
     c = xp.add(a, b)
     d = partial_reduce(c, np.sum, split_every={0: 3})
 
-    # specify max_total_num_input_blocks to force d to fuse
-    opt_fn = fuse_multiple_levels(max_total_num_input_blocks=6)
+    opt_fn = fuse_multiple_levels()
 
     d.visualize(optimize_function=opt_fn)
 
@@ -1176,7 +1198,7 @@ def test_optimize_stack(spec):
     c = xp.stack((a, b), axis=0)
     d = c + 1
     # try to fuse all ops into one (d will fuse with c, but c won't fuse with a and b)
-    d.compute(optimize_function=fuse_multiple_levels(max_total_num_input_blocks=10))
+    d.compute(optimize_function=fuse_multiple_levels())
 
 
 def test_optimize_concat(spec):
@@ -1186,4 +1208,4 @@ def test_optimize_concat(spec):
     c = xp.concat((a, b), axis=0)
     d = c + 1
     # try to fuse all ops into one (d will fuse with c, but c won't fuse with a and b)
-    d.compute(optimize_function=fuse_multiple_levels(max_total_num_input_blocks=10))
+    d.compute(optimize_function=fuse_multiple_levels())