kubeadm Configuration (v1beta3)

Overview

Package v1beta3 defines the v1beta3 version of the kubeadm configuration file format. This version improves on the v1beta2 format by fixing some minor issues and adding a few new fields.

A list of changes since v1beta2:

• The deprecated "ClusterConfiguration.useHyperKubeImage" field has been removed. Kubeadm no longer supports the hyperkube image.
• The "ClusterConfiguration.DNS.Type" field has been removed since CoreDNS is the only supported DNS server type by kubeadm.
• Include "datapolicy" tags on the fields that hold secrets. This would result in the field values to be omitted when API structures are printed with klog.
• Add "InitConfiguration.SkipPhases", "JoinConfiguration.SkipPhases" to allow skipping a list of phases during kubeadm init/join command execution.
• Add "InitConfiguration.NodeRegistration.ImagePullPolicy" and "JoinConfiguration.NodeRegistration.ImagePullPolicy" to allow specifying the images pull policy during kubeadm "init" and "join". The value must be one of "Always", "Never" or "IfNotPresent". "IfNotPresent" is the default, which has been the existing behavior prior to this addition.
• Add "InitConfiguration.Patches.Directory", "JoinConfiguration.Patches.Directory" to allow the user to configure a directory from which to take patches for components deployed by kubeadm.
• Move the BootstrapToken* API and related utilities out of the "kubeadm" API group to a new group "bootstraptoken". The kubeadm API version v1beta3 no longer contains the BootstrapToken* structures.

Migration from old kubeadm config versions

• kubeadm v1.15.x and newer can be used to migrate from v1beta1 to v1beta2.
• kubeadm v1.22.x and newer no longer support v1beta1 and older APIs, but can be used to migrate v1beta2 to v1beta3.
• kubeadm v1.27.x and newer no longer support v1beta2 and older APIs,

Basics

The preferred way to configure kubeadm is to pass an YAML configuration file with the --config option. Some of the configuration options defined in the kubeadm config file are also available as command line flags, but only the most common/simple use case are supported with this approach.

A kubeadm config file could contain multiple configuration types separated using three dashes (---).

kubeadm supports the following configuration types:

apiVersion: kubeadm.k8s.io/v1beta3
kind: InitConfiguration

apiVersion: kubeadm.k8s.io/v1beta3
kind: ClusterConfiguration

apiVersion: kubelet.config.k8s.io/v1beta1
kind: KubeletConfiguration

apiVersion: kubeproxy.config.k8s.io/v1alpha1
kind: KubeProxyConfiguration

apiVersion: kubeadm.k8s.io/v1beta3
kind: JoinConfiguration

To print the defaults for "init" and "join" actions use the following commands:

kubeadm config print init-defaults
kubeadm config print join-defaults

The list of configuration types that must be included in a configuration file depends by the action you are performing (init or join) and by the configuration options you are going to use (defaults or advanced customization).

If some configuration types are not provided, or provided only partially, kubeadm will use default values; defaults provided by kubeadm includes also enforcing consistency of values across components when required (e.g. --cluster-cidr flag on controller manager and clusterCIDR on kube-proxy).

Users are always allowed to override default values, with the only exception of a small subset of setting with relevance for security (e.g. enforce authorization-mode Node and RBAC on api server).

If the user provides a configuration types that is not expected for the action you are performing, kubeadm will ignore those types and print a warning.

Kubeadm init configuration types

When executing kubeadm init with the --config option, the following configuration types could be used: InitConfiguration, ClusterConfiguration, KubeProxyConfiguration, KubeletConfiguration, but only one between InitConfiguration and ClusterConfiguration is mandatory.

apiVersion: kubeadm.k8s.io/v1beta3
kind: InitConfiguration
bootstrapTokens:
  ...
nodeRegistration:
  ...

The InitConfiguration type should be used to configure runtime settings, that in case of kubeadm init are the configuration of the bootstrap token and all the setting which are specific to the node where kubeadm is executed, including:

• NodeRegistration, that holds fields that relate to registering the new node to the cluster; use it to customize the node name, the CRI socket to use or any other settings that should apply to this node only (e.g. the node ip).

• LocalAPIEndpoint, that represents the endpoint of the instance of the API server to be deployed on this node; use it e.g. to customize the API server advertise address.

apiVersion: kubeadm.k8s.io/v1beta3
kind: ClusterConfiguration
networking:
  ...
etcd:
  ...
apiServer:
  extraArgs:
    ...
  extraVolumes:
    ...
...

The ClusterConfiguration type should be used to configure cluster-wide settings, including settings for:

• networking that holds configuration for the networking topology of the cluster; use it e.g. to customize Pod subnet or services subnet.

• etcd: use it e.g. to customize the local etcd or to configure the API server for using an external etcd cluster.

• kube-apiserver, kube-scheduler, kube-controller-manager configurations; use it to customize control-plane components by adding customized setting or overriding kubeadm default settings.

apiVersion: kubeproxy.config.k8s.io/v1alpha1
kind: KubeProxyConfiguration
  ...

The KubeProxyConfiguration type should be used to change the configuration passed to kube-proxy instances deployed in the cluster. If this object is not provided or provided only partially, kubeadm applies defaults.

See https://kubernetes.io/docs/reference/command-line-tools-reference/kube-proxy/ or https://pkg.go.dev/k8s.io/kube-proxy/config/v1alpha1#KubeProxyConfiguration for kube-proxy official documentation.

apiVersion: kubelet.config.k8s.io/v1beta1
kind: KubeletConfiguration
  ...

The KubeletConfiguration type should be used to change the configurations that will be passed to all kubelet instances deployed in the cluster. If this object is not provided or provided only partially, kubeadm applies defaults.

See https://kubernetes.io/docs/reference/command-line-tools-reference/kubelet/ or https://pkg.go.dev/k8s.io/kubelet/config/v1beta1#KubeletConfiguration for kubelet official documentation.

Here is a fully populated example of a single YAML file containing multiple configuration types to be used during a kubeadm init run.

apiVersion: kubeadm.k8s.io/v1beta3
kind: InitConfiguration
bootstrapTokens:
  - token: "9a08jv.c0izixklcxtmnze7"
    description: "kubeadm bootstrap token"
    ttl: "24h"
  - token: "783bde.3f89s0fje9f38fhf"
    description: "another bootstrap token"
    usages:
      - authentication
      - signing
    groups:
      - system:bootstrappers:kubeadm:default-node-token
nodeRegistration:
  name: "ec2-10-100-0-1"
  criSocket: "/var/run/dockershim.sock"
  taints:
    - key: "kubeadmNode"
      value: "someValue"
      effect: "NoSchedule"
  kubeletExtraArgs:
    v: 4
  ignorePreflightErrors:
    - IsPrivilegedUser
  imagePullPolicy: "IfNotPresent"
localAPIEndpoint:
  advertiseAddress: "10.100.0.1"
  bindPort: 6443
certificateKey: "e6a2eb8581237ab72a4f494f30285ec12a9694d750b9785706a83bfcbbbd2204"
skipPhases:
  - addon/kube-proxy
---
apiVersion: kubeadm.k8s.io/v1beta3
kind: ClusterConfiguration
etcd:
  # one of local or external
  local:
    imageRepository: "registry.k8s.io"
    imageTag: "3.2.24"
    dataDir: "/var/lib/etcd"
    extraArgs:
      listen-client-urls: "http://10.100.0.1:2379"
    serverCertSANs:
      -  "ec2-10-100-0-1.compute-1.amazonaws.com"
    peerCertSANs:
      - "10.100.0.1"
  # external:
    # endpoints:
    # - "10.100.0.1:2379"
    # - "10.100.0.2:2379"
    # caFile: "/etcd/kubernetes/pki/etcd/etcd-ca.crt"
    # certFile: "/etcd/kubernetes/pki/etcd/etcd.crt"
    # keyFile: "/etcd/kubernetes/pki/etcd/etcd.key"
networking:
  serviceSubnet: "10.96.0.0/16"
  podSubnet: "10.244.0.0/24"
  dnsDomain: "cluster.local"
kubernetesVersion: "v1.21.0"
controlPlaneEndpoint: "10.100.0.1:6443"
apiServer:
  extraArgs:
    authorization-mode: "Node,RBAC"
  extraVolumes:
    - name: "some-volume"
      hostPath: "/etc/some-path"
      mountPath: "/etc/some-pod-path"
      readOnly: false
      pathType: File
  certSANs:
    - "10.100.1.1"
    - "ec2-10-100-0-1.compute-1.amazonaws.com"
  timeoutForControlPlane: 4m0s
controllerManager:
  extraArgs:
    "node-cidr-mask-size": "20"
  extraVolumes:
    - name: "some-volume"
      hostPath: "/etc/some-path"
      mountPath: "/etc/some-pod-path"
      readOnly: false
      pathType: File
scheduler:
  extraArgs:
    bind-address: "10.100.0.1"
  extraVolumes:
    - name: "some-volume"
      hostPath: "/etc/some-path"
      mountPath: "/etc/some-pod-path"
      readOnly: false
      pathType: File
certificatesDir: "/etc/kubernetes/pki"
imageRepository: "registry.k8s.io"
clusterName: "example-cluster"
---
apiVersion: kubelet.config.k8s.io/v1beta1
kind: KubeletConfiguration
# kubelet specific options here
---
apiVersion: kubeproxy.config.k8s.io/v1alpha1
kind: KubeProxyConfiguration
# kube-proxy specific options here

Kubeadm join configuration types

When executing kubeadm join with the --config option, the JoinConfiguration type should be provided.

apiVersion: kubeadm.k8s.io/v1beta3
kind: JoinConfiguration
  ...

The JoinConfiguration type should be used to configure runtime settings, that in case of kubeadm join are the discovery method used for accessing the cluster info and all the setting which are specific to the node where kubeadm is executed, including:

• nodeRegistration, that holds fields that relate to registering the new node to the cluster; use it to customize the node name, the CRI socket to use or any other settings that should apply to this node only (e.g. the node ip).

• apiEndpoint, that represents the endpoint of the instance of the API server to be eventually deployed on this node.

Resource Types

• ClusterConfiguration
• InitConfiguration
• JoinConfiguration

BootstrapToken

Appears in:

• InitConfiguration

BootstrapToken describes one bootstrap token, stored as a Secret in the cluster

Field	Description
token [Required] BootstrapTokenString	token is used for establishing bidirectional trust between nodes and control-planes. Used for joining nodes in the cluster.
description string	description sets a human-friendly message why this token exists and what it's used for, so other administrators can know its purpose.
ttl meta/v1.Duration	ttl defines the time to live for this token. Defaults to 24h. expires and ttl are mutually exclusive.
expires meta/v1.Time	expires specifies the timestamp when this token expires. Defaults to being set dynamically at runtime based on the ttl. expires and ttl are mutually exclusive.
usages []string	usages describes the ways in which this token can be used. Can by default be used for establishing bidirectional trust, but that can be changed here.
groups []string	groups specifies the extra groups that this token will authenticate as when/if used for authentication

BootstrapTokenString

Appears in:

• BootstrapToken

BootstrapTokenString is a token of the format abcdef.abcdef0123456789 that is used for both validation of the practically of the API server from a joining node's point of view and as an authentication method for the node in the bootstrap phase of "kubeadm join". This token is and should be short-lived.

Field	Description
- [Required] string	No description provided.
- [Required] string	No description provided.

ClusterConfiguration

ClusterConfiguration contains cluster-wide configuration for a kubeadm cluster

Field	Description
apiVersion string	kubeadm.k8s.io/v1beta3
kind string	ClusterConfiguration
etcd Etcd	etcd holds the configuration for etcd.
networking Networking	networking holds configuration for the networking topology of the cluster.
kubernetesVersion string	kubernetesVersion is the target version of the control plane.
controlPlaneEndpoint string	controlPlaneEndpoint sets a stable IP address or DNS name for the control plane. It can be a valid IP address or a RFC-1123 DNS subdomain, both with optional TCP port. In case the controlPlaneEndpoint is not specified, the advertiseAddress + bindPort are used; in case the controlPlaneEndpoint is specified but without a TCP port, the bindPort is used. Possible usages are: In a cluster with more than one control plane instances, this field should be assigned the address of the external load balancer in front of the control plane instances. In environments with enforced node recycling, the controlPlaneEndpoint could be used for assigning a stable DNS to the control plane.
apiServer APIServer	apiServer contains extra settings for the API server.
controllerManager ControlPlaneComponent	controllerManager contains extra settings for the controller manager.
scheduler ControlPlaneComponent	scheduler contains extra settings for the scheduler.
dns DNS	dns defines the options for the DNS add-on installed in the cluster.
certificatesDir string	certificatesDir specifies where to store or look for all required certificates.
imageRepository string	imageRepository sets the container registry to pull images from. If empty, registry.k8s.io will be used by default. In case of kubernetes version is a CI build (kubernetes version starts with ci/) gcr.io/k8s-staging-ci-images will be used as a default for control plane components and for kube-proxy, while registry.k8s.io will be used for all the other images.
featureGates map[string]bool	featureGates contains the feature gates enabled by the user.
clusterName string	The cluster name.

InitConfiguration

InitConfiguration contains a list of elements that is specific "kubeadm init"-only runtime information. kubeadm init-only information. These fields are solely used the first time kubeadm init runs. After that, the information in the fields IS NOT uploaded to the kubeadm-config ConfigMap that is used by kubeadm upgrade for instance. These fields must be omitempty.

Field	Description
apiVersion string	kubeadm.k8s.io/v1beta3
kind string	InitConfiguration
bootstrapTokens []BootstrapToken	bootstrapTokens is respected at kubeadm init time and describes a set of Bootstrap Tokens to create. This information IS NOT uploaded to the kubeadm cluster configmap, partly because of its sensitive nature
nodeRegistration NodeRegistrationOptions	nodeRegistration holds fields that relate to registering the new control-plane node to the cluster.
localAPIEndpoint APIEndpoint	localAPIEndpoint represents the endpoint of the API server instance that's deployed on this control plane node. In HA setups, this differs from ClusterConfiguration.controlPlaneEndpoint in the sense that controlPlaneEndpoint is the global endpoint for the cluster, which then load-balances the requests to each individual API server. This configuration object lets you customize what IP/DNS name and port the local API server advertises it's accessible on. By default, kubeadm tries to auto-detect the IP of the default interface and use that, but in case that process fails you may set the desired value here.
certificateKey string	certificateKey sets the key with which certificates and keys are encrypted prior to being uploaded in a Secret in the cluster during the uploadcerts init phase.
skipPhases []string	skipPhases is a list of phases to skip during command execution. The list of phases can be obtained with the kubeadm init --help command. The flag "--skip-phases" takes precedence over this field.
patches Patches	patches contains options related to applying patches to components deployed by kubeadm during kubeadm init.

JoinConfiguration

JoinConfiguration contains elements describing a particular node.

Field	Description
apiVersion string	kubeadm.k8s.io/v1beta3
kind string	JoinConfiguration
nodeRegistration NodeRegistrationOptions	nodeRegistration holds fields that relate to registering the new control-plane node to the cluster.
caCertPath string	caCertPath is the path to the SSL certificate authority used to secure comunications between a node and the control-plane. Defaults to "/etc/kubernetes/pki/ca.crt".
discovery [Required] Discovery	discovery specifies the options for the kubelet to use during the TLS bootstrap process.
controlPlane JoinControlPlane	controlPlane defines the additional control plane instance to be deployed on the joining node. If nil, no additional control plane instance will be deployed.
skipPhases []string	skipPhases is a list of phases to skip during command execution. The list of phases can be obtained with the kubeadm join --help command. The flag --skip-phases takes precedence over this field.
patches Patches	patches contains options related to applying patches to components deployed by kubeadm during kubeadm join.

APIEndpoint

Appears in:

• InitConfiguration

• JoinControlPlane

APIEndpoint struct contains elements of API server instance deployed on a node.

Field	Description
advertiseAddress string	advertiseAddress sets the IP address for the API server to advertise.
bindPort int32	bindPort sets the secure port for the API Server to bind to. Defaults to 6443.

APIServer

Appears in:

• ClusterConfiguration

APIServer holds settings necessary for API server deployments in the cluster

Field	Description
ControlPlaneComponent [Required] ControlPlaneComponent	(Members of ControlPlaneComponent are embedded into this type.) No description provided.
certSANs []string	certSANs sets extra Subject Alternative Names (SANs) for the API Server signing certificate.
timeoutForControlPlane meta/v1.Duration	timeoutForControlPlane controls the timeout that we wait for API server to appear.

BootstrapTokenDiscovery

Appears in:

• Discovery

BootstrapTokenDiscovery is used to set the options for bootstrap token based discovery

Field	Description
token [Required] string	token is a token used to validate cluster information fetched from the control-plane.
apiServerEndpoint string	apiServerEndpoint is an IP or domain name to the API server from which information will be fetched.
caCertHashes []string	caCertHashes specifies a set of public key pins to verify when token-based discovery is used. The root CA found during discovery must match one of these values. Specifying an empty set disables root CA pinning, which can be unsafe. Each hash is specified as <type>:<value>, where the only currently supported type is "sha256". This is a hex-encoded SHA-256 hash of the Subject Public Key Info (SPKI) object in DER-encoded ASN.1. These hashes can be calculated using, for example, OpenSSL.
unsafeSkipCAVerification bool	unsafeSkipCAVerification allows token-based discovery without CA verification via caCertHashes. This can weaken the security of kubeadm since other nodes can impersonate the control-plane.

ControlPlaneComponent

Appears in:

• ClusterConfiguration

• APIServer

ControlPlaneComponent holds settings common to control plane component of the cluster

Field	Description
extraArgs map[string]string	extraArgs is an extra set of flags to pass to the control plane component. A key in this map is the flag name as it appears on the command line except without leading dash(es).
extraVolumes []HostPathMount	extraVolumes is an extra set of host volumes, mounted to the control plane component.

DNS

Appears in:

• ClusterConfiguration

DNS defines the DNS addon that should be used in the cluster

Field	Description
ImageMeta [Required] ImageMeta	(Members of ImageMeta are embedded into this type.) imageMeta allows to customize the image used for the DNS component.

Discovery

Appears in:

• JoinConfiguration

Discovery specifies the options for the kubelet to use during the TLS Bootstrap process.

Field	Description
bootstrapToken BootstrapTokenDiscovery	bootstrapToken is used to set the options for bootstrap token based discovery. bootstrapToken and file are mutually exclusive.
file FileDiscovery	file is used to specify a file or URL to a kubeconfig file from which to load cluster information. bootstrapToken and file are mutually exclusive.
tlsBootstrapToken string	tlsBootstrapToken is a token used for TLS bootstrapping. If bootstrapToken is set, this field is defaulted to .bootstrapToken.token, but can be overridden. If file is set, this field must be set in case the KubeConfigFile does not contain any other authentication information
timeout meta/v1.Duration	timeout modifies the discovery timeout.

Etcd

Appears in:

• ClusterConfiguration

Etcd contains elements describing Etcd configuration.

Field	Description
local LocalEtcd	local provides configuration knobs for configuring the local etcd instance. local and external are mutually exclusive.
external ExternalEtcd	external describes how to connect to an external etcd cluster. local and external are mutually exclusive.

ExternalEtcd

Appears in:

• Etcd

ExternalEtcd describes an external etcd cluster. Kubeadm has no knowledge of where certificate files live and they must be supplied.

Field	Description
endpoints [Required] []string	endpoints contains the list of etcd members.
caFile [Required] string	caFile is an SSL Certificate Authority (CA) file used to secure etcd communication. Required if using a TLS connection.
certFile [Required] string	certFile is an SSL certification file used to secure etcd communication. Required if using a TLS connection.
keyFile [Required] string	keyFile is an SSL key file used to secure etcd communication. Required if using a TLS connection.

FileDiscovery

Appears in:

• Discovery

FileDiscovery is used to specify a file or URL to a kubeconfig file from which to load cluster information.

Field	Description
kubeConfigPath [Required] string	kubeConfigPath is used to specify the actual file path or URL to the kubeconfig file from which to load cluster information.

HostPathMount

Appears in:

• ControlPlaneComponent

HostPathMount contains elements describing volumes that are mounted from the host.

Field	Description
name [Required] string	name is the name of the volume inside the Pod template.
hostPath [Required] string	hostPath is the path in the host that will be mounted inside the Pod.
mountPath [Required] string	mountPath is the path inside the Pod where hostPath will be mounted.
readOnly bool	readOnly controls write access to the volume.
pathType core/v1.HostPathType	pathType is the type of the hostPath.

ImageMeta

Appears in:

• DNS

• LocalEtcd

ImageMeta allows to customize the image used for components that are not originated from the Kubernetes/Kubernetes release process

Field	Description
imageRepository string	imageRepository sets the container registry to pull images from. If not set, the imageRepository defined in ClusterConfiguration will be used instead.
imageTag string	imageTag allows to specify a tag for the image. In case this value is set, kubeadm does not change automatically the version of the above components during upgrades.

JoinControlPlane

Appears in:

• JoinConfiguration

JoinControlPlane contains elements describing an additional control plane instance to be deployed on the joining node.

Field	Description
localAPIEndpoint APIEndpoint	localAPIEndpoint represents the endpoint of the API server instance to be deployed on this node.
certificateKey string	certificateKey is the key that is used for decryption of certificates after they are downloaded from the secret upon joining a new control plane node. The corresponding encryption key is in the InitConfiguration.

LocalEtcd

Appears in:

• Etcd

LocalEtcd describes that kubeadm should run an etcd cluster locally

Field	Description
ImageMeta [Required] ImageMeta	(Members of ImageMeta are embedded into this type.) ImageMeta allows to customize the container used for etcd.
dataDir [Required] string	dataDir is the directory etcd will place its data. Defaults to "/var/lib/etcd".
extraArgs map[string]string	extraArgs are extra arguments provided to the etcd binary when run inside a static Pod. A key in this map is the flag name as it appears on the command line except without leading dash(es).
serverCertSANs []string	serverCertSANs sets extra Subject Alternative Names (SANs) for the etcd server signing certificate.
peerCertSANs []string	peerCertSANs sets extra Subject Alternative Names (SANs) for the etcd peer signing certificate.

Networking

Appears in:

• ClusterConfiguration

Networking contains elements describing cluster's networking configuration

Field	Description
serviceSubnet string	serviceSubnet is the subnet used by Kubernetes Services. Defaults to "10.96.0.0/12".
podSubnet string	podSubnet is the subnet used by Pods.
dnsDomain string	dnsDomain is the DNS domain used by Kubernetes Services. Defaults to "cluster.local".

NodeRegistrationOptions

Appears in:

• InitConfiguration

• JoinConfiguration

NodeRegistrationOptions holds fields that relate to registering a new control-plane or node to the cluster, either via "kubeadm init" or "kubeadm join"

Field	Description
name string	name is the .metadata.name field of the Node API object that will be created in this kubeadm init or kubeadm join operation. This field is also used in the CommonName field of the kubelet's client certificate to the API server. Defaults to the hostname of the node if not provided.
criSocket string	criSocket is used to retrieve container runtime info. This information will be annotated to the Node API object, for later re-use.
taints [Required] []core/v1.Taint	taints specifies the taints the Node API object should be registered with. If this field is unset, i.e. nil, it will be defaulted with a control-plane taint for control-plane nodes. If you don't want to taint your control-plane node, set this field to an empty list, i.e. taints: [] in the YAML file. This field is solely used for Node registration.
kubeletExtraArgs map[string]string	kubeletExtraArgs passes through extra arguments to the kubelet. The arguments here are passed to the kubelet command line via the environment file kubeadm writes at runtime for the kubelet to source. This overrides the generic base-level configuration in the kubelet-config ConfigMap. Flags have higher priority when parsing. These values are local and specific to the node kubeadm is executing on. A key in this map is the flag name as it appears on the command line except without leading dash(es).
ignorePreflightErrors []string	ignorePreflightErrors provides a list of pre-flight errors to be ignored when the current node is registered, e.g. IsPrevilegedUser,Swap. Value all ignores errors from all checks.
imagePullPolicy core/v1.PullPolicy	imagePullPolicy specifies the policy for image pulling during kubeadm "init" and "join" operations. The value of this field must be one of "Always", "IfNotPresent" or "Never". If this field is not set, kubeadm will default it to "IfNotPresent", or pull the required images if not present on the host.

Patches

Appears in:

• InitConfiguration

• JoinConfiguration

Patches contains options related to applying patches to components deployed by kubeadm.

Field

Description

directory string

directory is a path to a directory that contains files named "target[suffix][+patchtype].extension". For example, "kube-apiserver0+merge.yaml" or just "etcd.json". "target" can be one of "kube-apiserver", "kube-controller-manager", "kube-scheduler", "etcd". "patchtype" can be one of "strategic" "merge" or "json" and they match the patch formats supported by kubectl. The default "patchtype" is "strategic". "extension" must be either "json" or "yaml". "suffix" is an optional string that can be used to determine which patches are applied first alpha-numerically.